There are several churches that claim to be the Patriarchate of Antioch, most of them, ironically enough, are Catholic:
Greek Orthodox Patriarchate of Antioch-- that is, the one that you're talking about here
Syriac Orthodox Patriarchate of Antioch-- the portion of the historic Patriarchate of Antioch that rejected the Council of Chalcedon
Maronite Patriarchate of Antioch-- originally a small group that followed the monothelete heresy. Came into Communion with Rome during the Crusades. Any claim their patriarch has to historical succession to the See of Antioch is beyond tendentious.
Melkite Greek Catholic Patriarchate of Antioch, Alexandria, and Jeruslam-- the Catholic party from the schism of 1724
Syriac Catholic Patriarchate of Antioch-- A group that broke off from the (anti-Chalcedonian) Syriac Orthodox Patriarchate of Antioch and joined the Catholic Church in the 17th under the influence of French and Italian missionaries.
To say that "the Melkites never participated in the schism" is a very strange read on history. While it is true that the Patriarch of Antioch Peter III wrote to the Patriarch of Constantinople Michael Cerularius in an attempt to mediate between Rome and Constantinople, the gist of his message is that while the Latins are wrong about the filioque, azymes, etc. there is no need to press the issue at the current time. However, he died in 1051, three years before the schism between Rome and Constantinople. What information we have about the patriarchate of Antioch in the second half of the 11th century would seem to indicate that, unsurprisingly, the Patriarchate of Antioch was very close with Constantinople and indeed was not in communion with Rome-- the city of Antioch was, after all, part of the Byzantine Empire from 969-1078 and movement of clergy between the two patriarchates was constant. The most important writer of the Patriarchate of Antioch from this time, Nikon of the Black Mountain wrote quite a bit of polemic against the Latins. By the end of the 11th century, however, the situation is even more clear, since when the Crusaders conquered the coastal regions of Syria and Palestine they set up their own Latin hierarchies in Antioch and Jerusalem, displacing the native bishops. So certainly from the beginning of the 12th century we can say with certainty that Antioch was not in communion with Rome.