The blood of the divine in the veins of mortals

We need to really do our best to understand what is happening when we partake of the Eucharist.  A faithful approach to understanding any doctrine of the church includes an articulate and honest assessment of the relationship between God and his people, throughout old covenant history.
When we begin to seek an understanding of what is actually occurring during the celebration of the Eucharist, we need to see the Eucharist as part of the covenant relationship between God and his people. 
Going back to the Abrahamic covenant, we can see that the sign of circumcision was not simply a nice memorial to remind the Jews of who they were.  The sign of circumcision literally made the receiver of the covenant sign a part of the people of God.  Literally, actually, truly, genuinely a part of God's people.
When we look at the sacrificial system given by God to his people during the Mosaic covenant, the blood that was spilled in the context of the sacrifice for the covering of sins was not simply a nice picture to remind the Jews how God had forgiven them.  That spilled blood literally covered their sins.  Literally, actually, truly, genuinely covered their sins.
So when we come to the celebration of the Eucharist, now, in the New Covenant, we are partaking of the literal body and blood of Jesus Christ. Literally, actually, truly, genuinely the body and blood of Christ, not a nice little picture to remind us of Jesus' sacrifice.
The mistake we do not want to make comes in one of two categories.  One, that the partaking of the Eucharist is simply a memorial, a non-sacramental reminder of Jesus' death (baptistic theology) or two, that the Eucharist is understood as a scientifically verifiable physical change in material, the bread and wine physically transform into flesh and blood (Roman Catholic theology).
The historic and Orthodox understanding of the Eucharistic recognizes the literal change of the elements into the sacramental flesh and blood.  This is a literal change in a sacramental manner.
It's not a pretend reminder but it is also not a scientific ontological change.  It is something far greater that we can label and understand as sacro-covenantal.