The Byzantines furiously protested the idea of Charlemagne being the Western Emperor. They thought it was illegal due to the fact that Charlemagne was proclaimed since Irene was the sitting Emperor, even if the Pope saw her as ineligible to rule. Soon after the coronation of Charlemagne, the Byzantines and Franks instituted the Pax Nicephori, which recognized the Frankish Authority in the west (before this, the official policy was that the Franks ruled at the pleasure of the Emperor, even though Constantinople would never march into Francia) although it did not recognize the Franks title of Emperor. To my knowledge, the Byzantines never did recognize the Franks as the Western "Roman" Emperor, although later texts, such as the Alexiad, do indeed refer to the Holy Roman Emperors as "Basilius" (Greek for sovereign and the traditional title of the Byzantine Emperors), meaning they saw them as "Emperor," but not "Roman Emperor."