It is one of the more crude objections to the doctrine of the Real Presence, but it remains an objection nonetheless and as such must be answered. The crude objection I am referring to is the assertion that if the bread and wine of the Eucharist truly become the real Body of Christ, then this would subject our Lord's Body to the humiliation of being subject to the full rigors of the digestive process, meaning eventually that parts of our Lord's sacred Body would inevitably be passed through the bowels and ejected as bodily waste through the process of defecation. This belief was called Stercoranism. A difficult thought for a believer to entertain, but it was a real objection some lodged against Transubstantiation, and once you think about it, it does have a certain internal logic. If we confess that what we put into our mouth is in fact the real Body, Blood, Soul and Divinity of our Lord Jesus Christ, then it seems to be logical that the sacramental Body would undergo the same processes as all other food that goes into the mouth, including final ejection of parts of it into the toilet about eight hours later. Fortunately this gruesome speculation has no truth to it, as we will see in this article examining the origin and answers to the objections posed by the Stercoranists.
Origins of Stercoranism
Stercoranism traces its origin to the Carolingian Eucharistic Controversy during the reign of the Frankish King Charles the Bald (840-877), in which St. Paschasius Radbertus and the Benedictine Ratramnus of Corbie engaged in a heated dispute about the nature of the Real Presence of Christ in the Eucharist. Seeking to clarify the Church's doctrine on the matter for Charles the Bald, St. Paschasius composed a work called De corpora et sanguine Domini and presented it to King Charles in 844. Though St. Paschasius was correct in affirming the Church's traditional doctrine of a substantial change in the elements into Christ's Body, the imperfect development of sacramental theology in the 9th century led Paschasius to some ambiguities in differentiating between the carnal and sacramental modes of Christ's presence. This ambiguity prompted Charles the Bald to ask for a rebuttal from another eminent theologian, Ratramnus of Corbie, who responded by asserting that Christ's historical Body is not the same thing as his sacramental Body, which would be later developed by the heresiarch Berengar into the teaching that Christ's presence in the sacrament was merely spiritual. The controversies over the nature of Christ's Real Presence went on from 844 until the late 11th century, mainly in France.
During the time of the controversy, it was a common objection of the partisans of Berengar and Ratramnus to accuse their opponents who believed in a real change of substance of "Stercoranism", that if Christ is taken into the mouth in a literal fashion, He must also go down into the stomach, suffer digestion and be expelled from the Body in the form of human excrement during defecation. This was never a main argument of the Berengarians, and it should be noted that it was only an accusation - there is no evidence that anyone on any side of the controversy actually professed Stercoranism. It was thrown out more for its shock value and potential to insult than as a serious objection. With the multiple condemnations of Berengar in the 1070s, the opponents of the Real Presence melted away and the Stercoranist objection with them.
Martin Luther revived a form of the objection when he spoke of priests belching after receiving the Body of Christ and ejecting pieces of the Lord's Body in their spittle. It was not the same objection exactly, but again it had to do with the question of the ejection of the Body of Christ through natural processes. In Luther's case, as with Berengar, he used the image for its shock value.
Teaching on the Nature and Duration of Christ's Presence
At first glance, it seems as if the Stercoranist objection might have some merit to it since, as we said, it has a certain simplistic, internal logic to it. Let us begin with Catholic doctrine. It is absolutely certain that, after the consecration of the elements, the bread and wine become the Body and Blood of Christ in such a real and complete manner that no bread or wine remain, having been entirely replaced in their substance with the Body and Blood of Jesus Christ, an act the Church has called Transubstantiation.
Furthermore, it is of faith that Christ is not only present under the signs of bread and wine in their wholeness, but that the entire Christ is present in each particle of the host or drop of the Precious Blood, so that he who so much as receives but a particle of the consecrated host receives the true Body, Blood, Soul and Divinity of Christ as much as he who receives a larger piece. This was taught definitively at the Council of Trent:
"If any one denieth, that, in the venerable sacrament of the Eucharist, the whole Christ is contained under each species, and under every part of each species, when separated; let him be anathema." (Session XIII, Canon III)
Thus, the Body and Blood of Christ are not only present in the whole species, but in each particle or drop of the species. Yet Trent not only teaches the presence of the whole Christ in every particle, but also reminds us that this presence is an enduring one; it does not only arise during the act of reception of Holy Communion, as the Lutherans taught, but endures after the Mass and the reception of communion:
"If any one saith, that, after the consecration is completed, the body and blood of our Lord Jesus Christ are not in the admirable sacrament of the Eucharist, but (are there) only during the use, whilst it is being taken, and not either before or after; and that, in the hosts, or consecrated particles, which are reserved or which remain after communion, the true Body of the Lord remaineth not; let him be anathema." (Session XIII, Canon IV)
Therefore the whole Christ is present in every particle, and not only during the act of reception, but after the act of reception as well. How long after? Here the supporters of Berengar and Ratramnus would assert that Stercoranism becomes a necessity; if Christ is present in what goes down our throats and into our stomachs, and if we know that the presence of Christ abides in the host and its particles after reception of communion, and if furthermore we know from experience that of the foods we eat certain parts of them are passed out of our body through excrement, then it follows that every time we receive Holy Communion we risk profaning the Body of the Lord by defecating pieces of it out.
Sacramentum et Res, Sacramentum Tantum, Res Tantum
Fortunately, this disgusting scenario does not follow from the teachings of Trent or the doctrine of Transubstantiation. When speaking of the sacrament of the Eucharist, it is very common to emphasize the real change in the elements when arguing against those who do not believe Catholic teaching, and hence there tends to be an emphasis on the substance of the Body and Blood of Our Lord - the objective Presence of Christ through which we receive God's grace ex opere operato, if we are in a state of grace with the proper dispositions. In sacramental theology, this is the sacramentum et res, the mystical reality that underlies the sacrament.
Yet, we must remember that there is another aspect of the sacraments, the sacramentum tantum, which is the sacramental sign. Every sacramental reality (sacramentum et res) is concealed behind a sense-perceptible sign, a sacramentum tantum. The sacramentum tantum is the part of the sacrament that we physically interact with; this sacred sign veils an inner mystical reality, the sacramentum et res, with which we come into contact through the mediation of the sacramentum tantum. This is how God uses the sacramental sign to make available to us the bounty of spiritual graces that are proper to the sacraments (the res tantum, the inner grace).
It is necessary to recall that the Eucharist, as with every sacrament, is symbolic. It is not merely symbolic, however. A sacrament is nothing other than a sacred sign that effects what it signifies. The sacramental symbol becomes vivifying when the proper "form" or words are added to it, in the words of the Scholastics, ubi verbum ad elementum ("where word is added to the element"). The form is necessary to elevate the sign from a mere sign to a sacred sign, as St. Augustine teaches:
"Now you are clean through the word which I have spoken unto you. Why does He not say, You are clean through the baptism wherewith you have been washed, but
through the word which I have spoken unto you, save only that in the water also it is the word that cleanses? Take away the word, and the water is neither more nor less than water. The word is added to the element, and there results the Sacrament, as if itself also a kind of visible word." 
Elsewhere, St. Augustine will refer to sacraments as "useful signs", prefiguring the later definition of a sacrament as a sign that effects what it signifies.
The point is that for a valid sacrament to exist a sacramentum tantum, a sacramental sign, is necessary. Since the purpose of the sign is that humans are able to interact with it, tradition has taught that these signs must necessarily be sense perceptible. St. Thomas Aquinas sees this truth as flowing from the nature of man as a material being; because man is material and interacts with the world through his senses, it is fitting that the sacramental signs be material and able to be comprehended by the senses as well:
"It is part of man's nature to acquire knowledge of the intelligible from the sensible. But a sign is that by means of which one attains to the knowledge of something else. Consequently, since the sacred things which are signified by the sacraments, are the spiritual and intelligible goods by means of which man is sanctified, it follows that the sacramental signs consist in sensible things: just as in the Divine Scriptures spiritual things are set before us under the guise of things sensible. And hence it is that sensible things are required for the sacraments" 
So a fundamental part of any sacrament is the presence of a sense-perceptible sign. Accordingly, this means that if there is no sense perceptible sign, the sacrament cannot emerge; or, if the sacrament has emerged, the destruction of the sense-perceptible sign also causes the loss of the sacrament - the removal of the sacramentum tantum eliminates the sacramentum et res. The way to get rid of a sacrament is to get rid of the sacramental sign.
A perfect example of this is the practice for disposing of a consecrated host that has been spoiled or desecrated, for example, if it was given to an infirm person in a hospital who chewed part of it but was unable to swallow and spit it out. In such a case, the host is disposed of by being placed in water to dissolve until such a time that it is no longer discernible as bread. Then the water is dumped down the sacrarium. The sense-perceptible sign of bread has been destroyed, and therefore the sacramentum et res is no longer present.
Could there still be very small, perhaps microscopic particles of bread remaining? Most certainly there could be and probably are, but that is not the point; the point is they are no longer sense-perceptible, and thus cannot serve as a sacramentum tantum. Sense perceptibility is the key.
Answering the Stercoranist Objection
Similarly, when we receive our Lord into our Body, the sacred host is chewed in our mouth and then swallowed. The act of reception of Holy Communion, provided we are properly disposed, makes available to us all of the graces that God intends to communicate to us through the sacrament. We receive the true Body of Christ into our own body. But at some point, we know not when, through the natural process of breaking down the host, the host is no longer discernible to the senses as bread and the presence of Christ is no longer there. Some posit this happens very early, shortly after the host is chewed and swallowed, since we are no longer sense perceptive of any particular piece of something inside our stomach; others say it happens later. Regardless of when we say it happens (and nobody knows precisely when), the fact is that the very process of consuming the sacred host sooner or later destroys the sacramentum tantum, and well before the act of defecation would occur. Therefore, even if it were true that some infinitesimally small particles of the host are ejected from the body during defecation, these are not recognizable to the senses as bread and hence are not a sacramentum tantum, for the sacramental sign was destroyed in the act of consuming.
Therefore, the objection that belief in a change of elements would lead to Stercoranism is without foundation, the key being a proper understanding of the symbolic aspect of every sacrament and the necessity of a sacramentum tantum for the emergence of a sacramentum et res.
 St. Augustine, Tractates in John, 80:3
 St. Augustine, De Doctrina Christiana, III.9
 St. Thomas Aquinas, STh, III, Q. 60, Art. 4