Muslim Holy Roman Empire
There isn't really any lore besides this: General Abdul Rahman defeats the Franks in the Battle of Tours and just keeps going, conquering France, after the collapse of the Ummayids his descendents split from the Abbasids and formed the Rahman Empire, conquering most of Western Europe.
Under Muslim rule, Western Europe experiences the Muslim golden age as well. A lot of science and math and a lot of religious debates are held in cities like Elbaris and Toledo.
The relationship between the Empire's 3 main dynasties and the Catholic church is that of mutual assured security, the Muslim high nobility agree to protect the church and enforce its rulings and tribute as long as they tolerate muslim rule and agree to co-exist, due to this the Church is at the mercy of the Muslim nobility, but the Muslims are also equally at the mercy of Rome: primarily because 2/3s of the empire was still Roman Catholic and most of the Muslim base of support is in Andalusia, Paris and parts of northern Italy, allowing lower nobles who are still predominantly christian to rally support if the church chooses to revolt. This dynamic allows the Empire to persist for centuries.
Until, unfortunately, around the 1400s, when the Empire went into a full civil war where Italy and the German lands break away, the Catholic church no longer paying it's tribute and openly revolting and Faranca falls into chaos, where Joan of Arc manages to liberate most of France, only stopping at Andalusia itself, where its still predominantly Muslim to this day.
The map shows the empire at its peak around 1032 AD.