My understanding is . . . .
For appointments outside the Patriarchal territories, i.e. outside the old Ottoman Empire, the Synod sends to Rome the “terna,” a list of 3 names, i.e. first, second, third choices. Rome does a secret consultation and then appoints the candidate of their choice. It is often the first choice, but does not have to be any of the three. For appointments within the Patr. territories, the Synod sends one name and awaits Rome’s nihil obstat. If Rome objects the Synod chooses another name.
The Holy See approval: The Latin Church uses the Congregation for Bishops and the Eastern Churches use the Congregation for Eastern Churches.
Congregation for the (Oriental) Eastern Churches:
Cardinal Leonardo Sandri, Prefect of the Congregation for Eastern Churches
Archbishop Cyril Vasil (Slovak Catholic), SJ, Secretary of the Congregation for Eastern Churches
Mgr. Maurizio Malvestiti, Undersecretary of the Congregation
Patriarch Ignace Youssif III Younan (Syrian Catholic)
Patriarch Antonios Naguib (Coptic Catholic)
Major Archbishop Cardinal Lubomyr Husar, M.S.U. (Ukrainian Greek Catholilc)
Patriarch Gregorios III Laham (Melkite Greek Catholic)
Patriarchal Vicar Cardinal Nasrallah Pierre Sfeir (Maronite Catholic)
Patriarch Cardinal Emmanuel III Delly (Chaldean Catholic)
Major Archbishop Cardinal Mar Varkey Vithayathil, C.SS.R. (Syro-Malabar Catholic)
Patriarch Nersès Bédros XIX Tarmouni (Armenian Catholic)
Major Archbishop Baselios Cleemis Thottunkal (Syro-Malankar Catholic)
Major Archbishop Lucian Mureşan (Romanian Greek Catholic)