John Calvin (as opposed to Roman Catholicism and Martin Luther) maintained that the "real presence" of Christ is present in the Lord's Supper.
And yet, not the "carnal" or physical presence. As the writer of this article says,
He [Calvin] asserted that Christ is truly present in the sacrament, but that his presence is brought about through the agency of the Holy Spirit, uniting the believer with the body and blood of Christ to be fed spiritually. Calvin, in agreement with Zwingli, believed that Christ’s body is in heaven and that it therefore cannot be contained locally in the Eucharist, but he did not think that Zwingli did justice to Christ’s words of institution, to Paul’s teaching in 1 Corinthians 10:16, or to the ancient tradition of the church. He believed in a spiritual understanding of the Presence that, in his opinion, is no less real than the localized understanding affirmed by the Roman Catholic Church and by Luther.
The sacrament of the Lord's Supper is essential to one's walk with God. There is something mystical about this meal (mystical in the "having a spiritual reality" meaning of the word). There is something beautiful in partaking of communion.
The Bible clearly teaches the bodily ascension into heaven - the real, physical body of Christ is not here on earth. To accept the reality of the dual nature of Christ (fully man and fully God) gives us the possibility of the "real" spiritual presence of Christ in communion, but not the physical presence.
It is also interesting to note that for all their claims that Calvinists destroy the plain sense of the words of institution “this is”, ubiquitarians themselves destroy the plain sense of the passages concerning Christ’s bodily ascension into heaven, which is that the God man Jesus Christ moved locally from one place to another in which place his physical body resides. (Remember all His talk about how He was about to “go” to His Father and “come again”?) Either way, both sides are constrained to hold an unliteral understanding of one passage of Scripture or the other.
I've heard this issue brought up like a mantra...making it an issue of division, instead of the unity it was meant to bring. I like this paragraph:
In the end, one thing we all must confess is that we are dealing with a tremendous and unfathomable mystery here. Let us tremble before the majesty of our divine-human King and shudder at the thought of treating our fellow worshipers with contempt, lest we be guilty of despising His beloved children whom he feeds graciously with the wonderful substance of his true body and blood. He has granted to His Church the tremendous blessing of feeding on His flesh and blood that it might be united in him, not so that we could exalt ourselves over one another and tear his body apart like a bunch of ravenous heathen cannibals.