We don't know the mechanism. The Eastern Church agrees that there is change in substance but that the mechanism remains a mystery. In other words, we agree with the meaning of the term transubstantiation but reject that it explains the mode by which the change is made.
In that regard, nothing has changed in either Catholic or Orthodox Church doctrine. Kolo correctly points to the fact that the Latin Church, faced with Protestant challenge, found it necessary to express the mystery in more legalistic and clearly defined terms in order not to confuse it with Lutheran "consubstantiation."
The Eastern Church to this day accepts, but not necessarily uses, the term metaousiois (change in substance, the equivalent of transubstantiation), because it expresses what the Church taught all along.
Seems to me this would be an easy thing to come together on because in the end we both receive the same Christ -truly wholly and substantially His Body ,Blood, Soul and Divinity