|Pope Francis meeting Patriarch Bartholomew in Turkey Dec. 2014|
Not popes, but patriarchs. This answer is embedded in history which is where things always get interesting and make more sense. There were five ancient patriarchates: basically self-governing territories under a chief bishop and his synod. Those five were Rome, Constantinople, Alexandria, Antioch, and Jerusalem. Remember that distances were greater when the whole world operated without technology and on foot or horseback. It was hardly practical for a centralized office to handle every local decision about the universal church, especially as languages and cultural contexts of each diocese were quite different. The law of the church (canon law) wasn't even informally standardized until the Middle Ages. Bishops came together for universal councils in places like Ephesus and Chalcedon for rulings on controversial questions and to resolve major conflicts. But for the most part, the patriarchates ran their dioceses effectively.
The papacy's profile soared after Pope Leo I's reign in the fifth century. Two hundred years earlier, Irenaeus had affirmed Rome as a "more powerful principality" rooted in the martyrdoms of Peter and Paul in that city. Popes before Leo I had also seen the Roman bishop as holding "pastoral care of all the churches." But Pope Leo was the first to declare that the Bishop of Rome assumed the fullness of power conferred on Peter by Christ. To be in communion with Rome, therefore, is to be in communion with all bishops and churches who confess now, have confessed, or will confess the Catholic faith.
Tensions gradually arose between the Eastern patriarchs and Rome over matters of theology, liturgy, and church practice. Authority and governance became a flashpoint, culminating in the Great Schism between East and West in 1054. The Eastern church claimed the name Orthodox, viewing the See of Rome as a "papal church." Eastern and Western leaders excommunicated each other and their constituencies, a ban that wasn't lifted until the time of Pope Paul VI in the twentieth century. Nationhood advanced as a preferred political identity, and increased nationalization of the churches proliferated. Some Eastern patriarchs remained loyal to the Pope including the Armenian, Chaldean, Coptic, Maronite, Melkite, and West Syrian patriarchates. Over twenty unique Catholic rites exist in full communion with the Roman Catholic Church today. The rest allied with the Eastern and Oriental Orthodox patriarchs. They have not been in communion with Rome for almost a thousand years. The dialogue of East and West continues.
101 Questions and Answers on Eastern Catholic Churches - Edward Faulk (Mahwah, NJ: Paulist Press, 2007)
You Are Peter: An Orthodox Theologian's Reflection on the Exercise of Papal Primacy - Olivier Clement (Hyde Park, NY: New City Press, 2003)