In fact, all the major schisms in Christian history were partly caused by rivalry. The divisions which arose from the 5th century show rivalry for precedence between three main Sees of Christendom: Constantinople, Alexandria, and Antioch. The 11th century Great Schism between the See of Rome and the See of Constantinople was probably caused by a rift of estrangement in their respective cultures but was nevertheless a matter of rivalry.
The Catholic Church and the Orthodox Churches reached a reunion in two Councils: The Council of Lyons I (1274) and the Council of Florence (concluded in 1445). However, the reunion fell upon the collapse of Constantinople to the Muslim Ottomans in 1453 since the military aid promised by Catholics did not help the Greek Orthodox in defending the city.
If we look at the world from our human perspective then we will surely be disappointed by the continuous failures of us Christians to be like Christ.