Given my background, I've been curious about this question for a little while: which, if any, of the Oriental churches is most like the Byzantines in what is particular about them (theological emphases, culture, liturgics, etc.) in comparison to the other Oriental churches?
For a little while I had a hope for the possibility of a Byzantine rite OO church. But it would seem that that is not likely to come to pass anytime soon. I am considering this as a possible alternative.