How was Christian intermarriage with the Muslim Turks viewed by the Byzantines?

Question

While learning about the Battle of Manzikert (1071) I was surprised to hear that the emperor Romanos IV agreed in the aftermath to a political marriage by giving his daughter to be married to Sultan Alp Arslan's son. How would the Byzantine Eastern Orthodox Christians have viewed such a marriage? Is it safe to assume that such marriages of marrying off a Christian daughter resulted in their forced conversion to Islam, and that they were not allowed to remain Christian?

In reading about another similar proposition/offer not long after to Alexius I Comnenus, I read the following:

The Great Seljuk Sultan Malik Shah was not unhappy with the apparent disintegration of the Rum Sultanate. He still felt that suzerainty over all Seljuks was his by right and in 1090AD he made Alexius an extraordinary offer. In exchange for a peace treaty and a marriage alliance the Sultan would withdraw all Seljuk forces from Anatolia and restore all Byzantine lands lost since Manzikert. It may have been a tempting offer but Alexius refused. Publicly Alexius could not consent to a marriage between his Christian daughter and a Moslem; while politically, the Seljuks of Rum were a useful buffer between Byzantium and the much more powerful Great Seljuks. 1

This seems a bit more understandable in terms of his reticence to condone such a marriage, because the public perception of this surely would have been unacceptable to official Eastern Orthodox beliefs. I'm curious also how many such intermarriages happened where the daughter was given in marriage to a Muslim Turk (for political reasons) including under the later Ottoman Empire up until the end of the Byzantine Empire, but more so how even a single instance would have been viewed by the Christian Byzantine EO Church? This is as much a church history question as a general historical and cultural curiosity.

Answer 1

Just a partial answer here to one of the questions posed in the body (not in the title):

"Is it safe to assume that such marriages of marrying off a Christian daughter resulted in their forced conversion to Islam, and that they were not allowed to remain Christian?"

It was not always the case that the wife would convert to Islam, at least when the wife was a daughter of an emperor.

Theodora Kantakouzene, born c. 1330

Theodora was one of the three daughters of Emperor John VI Kantakouzenos by his wife Irene Asanina... In January 1346, to cement her father's alliance with the rising Ottoman emirate and to prevent the Ottomans from giving their aid to the Empress-regent Anna of Savoy during the ongoing civil war, she was betrothed to the Ottoman ruler, Orhan Gazi.

...Theodora remained a Christian after her marriage, and was active in supporting the Christians living under Ottoman rule.

I have nothing to offer on how other Byzantines viewed this arrangement, but as @Spencer comments, when the emperor supports it do other Byzantines matter?

She is known to have returned to the palace in Constantinople after her husband the Sultan's death in 1362. Last we know of her she is imprisoned by Andronikos IV (reigned just 1379–81) but this is more likely to be because he usurped his own father than another factor.