By presence does that mean believing that the wine has turned into the literal physical blood of Christ? That they are drinking the actual human corpuscles of Jesus?
I'm not sure that even those who believe in transubstantiation would phrase it this way, since I think they affirm that it's the substance (in the Aristotelian sense) that's transformed, and I would guess that individual red blood cells are accidents rather than substance. But I'll let the Catholics in this thread speak for themselves on this. I don't connect well with Aristotle's view of metaphysics, so I'm not the best one to describe it (or, by extension, St Thomas' explanations that build on Aristotle).
I will speak as best as I can from my own tradition. Starting with our liturgy, because that's where everything starts with us.
Liturgy:
From Eucharistic Prayer A:
"Recalling [Christ's] death, resurrection, and ascension, we offer you these gifts. Sanctify them by your Holy Spirit to be for your people the Body and Blood of your Son, the holy food and drink of new and unending life in him. Sanctify us also that we may faithfully receive this holy Sacrament, and serve you in unity, constancy, and peace; and at the last day bring us with all your saints into the joy of your eternal kingdom."
From Eucharistic Prayer D:
"Lord, we pray that in your goodness and mercy your Holy Spirit may descend upon us, and upon these gifts, sanctifying them and showing them to be holy gifts for your holy people, the bread of life and the cup of salvation, the Body and Blood of your Son Jesus Christ."
Note that we do not put leftover wine back in the bottle to be consumed as an ordinary beverage; similarly with leftover bread. Once they have been consecrated, they are either consumed or reverently reserved for later occasions of Communion.
Reflections:
God is spirit, so God's presence in the Sacrament is, at the very least, spiritual. Note, importantly, that what is spiritual is real; it's not just our imagination or memories. God is acting here, and God is present here, in a particular way that goes beyond God's generic omnipresence.
And God is present in the consecrated Elements. The Elements become connected in some way to the body and blood that Jesus inhabited while on earth and that were broken in the crucifixion.
At that point, I think, we reach the end of our understanding. Some of us make various guesses as to the underlying mechanism -- transubstantiation, or consubstantiation, or other explanations. Some of us simply say "God is here, and I don't know how."
The Anglican tradition is one that's comfortable with ambiguity, and with saying that there are limits to our understanding of God and God's activity. Here is one of the places where this shows up. Still, in our liturgical words and actions, we're affirming that God is doing something at the Altar, that something holy is happening here, that it's more than just Christians remembering what Jesus did for us in the past.
