Has there ever been an Eastern Catholic Pope (Post-Schism)? Every Pope that I've heard of has been from the Latin rite/the Roman Catholic Church. Is it possible for the Pope to be a bishop from an Eastern Catholic church instead? For example I reckon it'd be really cool to have a Pope from the Syro-Malabar Church or the Ethiopian or Eritrean Churches, or the Byzantine Melkite Church.

I'm wondering if this has happened at any time historically and if not, why not? Is there anything preventing an Eastern Catholic bishop from being elected Pope?

If an Eastern bishop were to be elected Pope, could they celebrate their native liturgical rite regularly in Vatican City? Or would they be forced into using the Roman Rite most of the time? My understanding is that this is possible because the Pope is considered omniritual: he can celebrate whatever liturgy he wants and is not tied down to the Latin Rite.

As a historical footnote, I'll point out that before the great schism there was a period of church history where the Popes were all imported from Byzantine Greece, Byzantine Syria, and Byzantine Sicily. However this was before the great schism and so is not as remarkable as it seems at first glance because Eastern and Western Christianity were united at the time. See here: Byzantine Papacy
There are really only two requirements for becoming pope: being male and being baptized into the Catholic Church.


  Canon 332 § 1 of the 1983 Code simply states that one already a bishop (nb: not necessarily a cardinal) who accepts legitimate papal election becomes pope immediately. One who is not yet a bishop (and the Church has elected several non-bishops to the papacy) can accept election, but must be immediately consecrated bishop. By implication, that would seem to require that a papabile (a) be male, and be willing (b) to be baptized, (c) ordained deacon, priest, and bishop, and (d) have the use of reason in order to accept election and, if necessary, holy orders. - In the Light of the Law 


Could an Eastern Rite cardinal be elected pope? Yes, it could happen and a pope is free to make changes to reflect his heritage, but we would have to wait and see if this ever happens.


  When Eastern Rite cardinals meet with other cardinals or con-celebrate the liturgy, they generally wear their own vestments. On the other hand, if a bi-ritual priest celebrates the liturgy, not native to him, he uses the other rite's vestments. Someone elected pope is the Bishop of Rome, so presumably he would celebrate henceforth as Bishop of Rome, i.e., Latin Rite. Since we haven't run into this situation in modern times (and probably don't have sufficient records when it did happen, which I think it did) they would probably have to make some new decisions about it, much like they did with Pope Benedict's resignation. 
  
  My guess is he would wear Western vestments when celebrating a Latin-rite Mass. As for his white cassock, I'm not sure. On the one hand that's kind of unique to the pope and in a class by itself, but then again it is based on Latin-rite dress. 
  
  Maybe the pope would like to retain some of his Eastern heritage. I don't know, I think that would be up to him. - Ask A Catholic


Update: Thanks to  @AthanasiusOfAlex and his comment: There seems to have been no Eastern Catholic pope. "Looking at the list of popes since 1054, the answer appears to be “no”."