Definitely primacy of honour. Keep in mind that before the schism over Chalcedon, there was no division, so rather than asking in terms of EO/OO, it might make more sense to ask in terms of how did Alexandria, Jerusalem, Antioch, later Constantinople, and other smaller bishoprics relate to Rome.
If you read Eusebius' Church History, the picture you see is Rome repeatedly trying to exercise greater authority, and soundly being shot down or ignored by everyone else. Even the bishop of Gaul, hardly a prominent see, rebuked Rome for excommunicating the bishops of Asia Minor for following their traditional Pascal practice. Alexandria was more often an arbitrator in theological disputes than Rome in the early Church.
Today, the Patriarchs of Alexandria, Ethiopia, India, Armenia, Syria, all cooperate as equals, as brothers. No one has felt the need to try to "replace" Rome as the bully of the Communion.