Dear Mexican,
In a word, the papacy.
The Armenian Apostolic Orthodox Church, for example, is so close to union with "Byzantine" Orthodoxy - but as came out during the published theological discussions with them is that they are afraid of losing their ecclesial identity if they did so.
The Armenian Church has a "college" of patriarchates with the Patriarch of Etchmiadzin (meaning "The descent of the Only-Begotten") being the first among equals. The survival of the Armenian people depends on the survival of their Holy Church.
There were Armenians who were formally "Byzantine Orthodox" and St Paissy Velichkovsky counted a number among his followers.
The Mechitarist Armenian Catholics have, on the other hand, achieved a profound esteem among their countrymen for their unquestioned commitment to the Armenian Church and people.
During the theological discussions between the Miaphysites and the Orthodox, it was shown that the only thing that separated those two "families of Orthodox Churches of the East" was one word - the definition of the Person/Natures of Christ.
Apart from that, the two families of Churches are identical in all else.
The same cannot be said of the Miaphysites and the RC Church.
However, the Miaphysites that I know don't let things like ecclesial divisions get in the way of good relationships!
Being at an Armenian Church once, I noticed that EVERY single car in the vast parking lot sported a Latin rosary on the rear-view mirror. When we had our petition to declare Pope John Paul II Day in Ontario, Armenian Orthodox were sending in signed copies . . .
We should pray that the Oriental and Eastern Orthodox Churches become "One, Holy, Orthodox, Catholic and Apostolic Church of the East" and then further ecumenical perspectives can be examined with the view to what the future might promise.
Alex