This is the Greek Catholic Patriarchate. However, I have been warned before that I mix churches up, a thing to be avoided. So from the Enc. Britt. I first quote: "an Eastern Catholic church of the Byzantine rite, in communion with Rome since the mid-19th century. A small body of Greek Catholics came into existence in Constantinople (modern Istanbul) and Thrace largely through the efforts of John Hyacinth Marango, a Latin priest, and Polycarp Anastasiadis, a Greek priest. An apostolic exarchate, eventually located in Athens, was created in 1911, and a separate exarchate at Istanbul was set up in 1932. The term Greek Catholic properly applies only to members of the Catholic church, either Byzantine or Latin rite, who are of Greek nationality, but Catholic Melchites and Ukrainians also call themselves Greek Catholics."
Then I take a closer look at a notice at the entrance I was wise enough to photograph, and see this is the Melkite Greek Catholic Patriarchate. That "changes the picture", because: (Enc. Britt. again): "[Melchite] also spelled Melkite, any of the Christians of Syria and Egypt who accepted the ruling of the Council of Chalcedon (451) affirming the two natures—divine and human—of Christ. Because they shared the theological position of the Byzantine emperor, they were derisively termed Melchites—that is, Royalists or Emperor's Men (from Syriac malkā: “king”)—by those who rejected the Chalcedonian definition and believed in only one nature in Christ (the Monophysite heresy). While the term originally referred only to Egyptian Christians, it came to be used for all Chalcedonians in the Middle East and finally, losing its pejorative tone, came to designate the faithful of the patriarchates of Alexandria, Jerusalem, and especially Antioch.
The Melchite community generally consisted of Greek colonists and the Arabicized populations of Egypt and Syria. They adopted the Byzantine rite and thus followed Michael Cerularius, patriarch of Constantinople, into schism with Rome in 1054. For several centuries afterward, the patriarch of Antioch attempted reunification with Rome, and a small number of Melchite Catholics emerged. Final union came in 1724, when Cyril VI, a Catholic, was elected patriarch of Antioch; he was followed by several bishops and a third of the faithful. The Orthodox who opposed union elected their own patriarch, Silvester, and obtained the legal recognition from the Ottoman government that assured them autonomy. About 100 years later, after much persecution and religious difficulties with Jesuits and Lebanese Maronites, the Catholics also received autonomous status from the Ottoman Turks, which allowed for normal activity and growth.
While there had been some few conversions to Catholicism in the patriarchates of Alexandria and Jerusalem, there is only one Catholic Melchite “patriarch of Antioch, Alexandria, Jerusalem and all the East.” In each patriarchate he has his own diocese (Damascus, Jerusalem, Alexandria) and is helped by a patriarchal vicar. There are seven archdioceses—Aleppo, Homs, and Latakia (all in Syria), Beirut and Tyre (both in Lebanon), Basra (in Iraq), and Petra-Philadelphia (Jordan). There are six dioceses, in Acre (Israel) and Baalbek, Baniyas, Saïda, Tripolis, and Zahleh-Furzol (all in Lebanon). The number of Catholic Melchites, who observe the Byzantine liturgy in their vernacular Arabic, totals about 250,000 with an additional 150,000 abroad, mainly in Brazil, Argentina, the United States, and Canada."