There's significant religious tension between Muslims and Christians during this period. But considering the rise of universities in Europe at a time when Islam was home to most advanced centers of learning, (i.e. Baghdad, Antioch, etc.) to what extent did they collaborate academically or scientifically, putting religious differences aside?
I don't know much about cases of real, simultaneous collaboration in the sense of visiting faculty and the like, but Western Christian academics certainly made use of Islamic sources, with two Muslim academics, Averroes (Ibn Rushd) and Avicenna (Ibn Sina) reaching almost canonical status within Western Christian academia. Some other Muslim scholars, such Rhazes (Muhammed ibn Zakariya al-Razi) also achieved prominence.
I am not familiar with the extent to which Muslim academics took an interest in Christian (whether Western or Eastern Christian) sources, or the extent to which the Byzantines took an interest in Muslim (or Western Christian) sources.