I have continued to get very good input honing in on the source of the issue from several priests and a bishop. I think I am finally beginning to better understand. This portion may be the most helpful ...
(From Fr. John)
Since it is His body, and outwardly bread, when we eat it the actual Body of Jesus Christ is united to our body and transfigures us. We literally are united, in a real, physical way, to the Body of Jesus Christ. This is not hocus pocus. It is the action of God to take what is natural and elevate it into reality, and thereby change us as well. HOW the Holy Spirit effects the change doesn't matter. That is what the entire argument about transubstantiation is actually about. If the term is used to simply mean that the gifts are changed into the Body and Blood of Christ, then fine, we agree. If it is used to mean that we accept the Aristotelian philosophical underpinnings of that notion, then we don't accept it. But note why. We don't accept it, not necessarily because we think it untrue, but because we cannot say with certainty how God changes the gifts. Now, the be fair, the Catholic Church no longer requires everyone to believe in the philosophical underpinnings of the word transubstantiation, but the word has taken to mean that the gifts are changed into the Body and Blood in a real way. Because of that, you have to be careful and ask what the Catholic means by using the word.
To be perfectly honest, I wasn't seeing the real substantial reason for disagreement either, which is why I stepped back to say "this is what we believe, if you believe this, then we agree". I see now the question is one of WHY we believe. If "transubstantiation" in one's definition includes the Aristotelian philosophizing, then we reject it. If it means simply that the bread and wine are changed into the Body and Blood of Our Lord by the Holy Spirit, then in essence we agree, as long as everything beyond that is Mystery. I note that Fr said that Catholicism no longer requires Catholics to agree to all of that, so this may well represent one of those cases where Rome has moved closer to Orthodoxy.
I for one am not looking/hoping for differences. I apologize if it has ever seemed that way. If we are the same on something, glory to God! I say that with Catholics, Baptists, Lutherans, Pentecostals, whoever. Where we are different, I seek to understand. But I think we do a disservice to either inflate or to minimize our differences when we talk about the Church. When we deal with each other as individuals, I much prefer to build on our common beliefs. It depends on exactly what we are discussing.
It seems to me that really, in terms of the Eucharist (not counting a few other things like leavened vs. unleavened, and some lesser points) ... in the very basic question of whether we agree on what the Eucharist IS, the answer is "maybe" ... and it depends on the theology behind the theology in the case of our Latin brother.
I hope this helps clarify a bit?