Elliott Holley
How did the Muslims in Eastern Roman (Byzantine) Constantinople live and interact with the locals?
Islam existed in the Byzantine Empire for many centuries. There is evidence of mosques in the empire as early as the 10th century, and there was at least one mosque in Constantinople in the 12th century, the period of the Crusades.

A modern image of Istanbul. The Ottoman mosques were actually not the first to exist in the city.
There is a lovely story, that when a group of western Crusaders tried to make trouble at the mosque, the inhabitants of Constantinople formed a protective barrier around the mosque, and protected their fellow citizens.
As this story illustrates, Constantinople had always been a cosmopolitan place, and despite what some people/Islamophobic bigots would have you believe, there was a degree of interaction, trade and normal life.
The state religion of Byzantium was Greek Orthodox Christianity, and that was an important part of the state and the culture. Most Muslims in the empire would probably have been merchants, traders and the like, just as people from many nations could be found in the capital, from as far afield as Russia and China.
In the 12th century, the emperor Manuel I Komnenos attempted to get the church to change the conversion process to make it easier for Muslims to convert to Greek Orthodox Christianity, but the clerics wouldn’t have it. Specifically, Manuel wanted to remove the part where Muslims were required to explicitly reject “the god of Muhammad”.
Since Christians and Muslims worship the same god, Manuel was in the right. But the church clerics were too inflexible, and they wouldn’t listen.
The Byzantines had a lot more in common with the civilised diplomacy and court life of neighbouring Islamic societies such as the Abbasid empire, and later the various Seljuk states, than they did with the somewhat “uncivilised” nations of the west, as least in the 11th and 12th century. Byzantine nobles could and often did move back and forth between the Seljuk court and Constantinople, and there was intermarriage between the two.

From the Islamic side, as is well known Islam permits freedom of worship to Christians and Jews etc, in exchange for the jizya tax, which was typically set at a level lower than what people had paid previously to the Byzantine government.
In the final centuries of the empire, the saying “rather the Sultan’s turban than the Pope’s tiara” became popular among the Byzantines. This reflected the general view of most people that it was preferable to be ruled by the Ottoman Turks (freedom of worship) than it was to be ruled by Catholic powers such as Venice, which tended to restrict religious freedom and attempt to force the much-disliked western Catholicism on the people.
Under the Ottoman rule, the millet system meant that Greek Orthodox inhabitants of the Ottoman Empire were able to keep the Greek Orthodox church, and were also judged under the rules of their own community.
The relationship of Islam to the Byzantine Empire was thus pretty complex, as relations between different societies and religions often are. There was everything from friendship, diplomatic alliances and marriages, to outright warfare between the empire and neighbouring Islamic powers, just as there was the same between the empire and neighbouring Catholic powers.
The idea of “Christianity” versus “Islam” is a myth, pushed by modern people with an agenda. It has absolutely nothing to do with actual history. In reality, it would be more accurate to speak of relations between the Greek Orthodox Byzantine Empire, the Catholic “west”, and the Sunni Muslim and Shia Muslim powers, as well as smaller players such as the Armenian church (which had its own complex relationship with Byzantium, including both wars and cooperation, at different times).
People like these overly simplistic, knee-jerk categorisations of things into “good and bad”, but reality just isn’t like that, and it’s especially important to remember that nuance, given modern society’s collective neuroses and the sheer level of disinformation and ignorance about anything to do with Islam.