You have asked two different questions. First, you ask about religious faith. Then you ask about the religious context. These are importantly different, and that you conflate them leads me to suspect you are only thinking of a subset of religions. Not all religions have scriptures. Not all religions demand faith. Not all religions are committed to an orthodoxy. So one can operate within a religious context without faith.
But even if we limit ourselves to the subset of Mediterranean religions that you apparently have in mind, you still seem to assume that members of a religion are bound to the philosophical viewpoints that previous members of that religion think they found in their scriptures. I don’t think this is true, and not just because there is a long history of religious revolution and reformation.
Immanuel Kant’s Religion within the Limits of Bare Reason represents one historically influential—and highly unorthodox—attempt to ground religion on reason without reference to revelation. The Jefferson Bible is another. And it was common—though not universal—for the early church fathers to declare that anything known to be true by revelation was provable by reason. Indeed, this was their guiding principle when trying to figure out which parts of the Bible to take literally and which to understand as being metaphorical (Biblical literalism being a thoroughly modern trend in Christianity).
Finally, we have to ask exactly what you mean by “compatible.” Kierkegaard is instructive here. In his view, the leap to faith is arational (i.e., not rational). But he insists that does not make it irrational (i.e., against reason). It happens separate from rationality, but need not represent a rejection of rationality. In that sense, the two could be compatible in the sense of both being unopposed parts of a single life even while not being compatible in the sense of being mutually reinforcing in all respects (leaving open the possibility that they may be mutually reinforcing in some respects, though they also may not be).
There are as many ways to be religious as there are people. Some people may reject reason. Others may embrace nothing but reason. And of course, plenty of people try their best to find a balance between the two (which is at least the right instinct, regardless of how successful they are). But in the end, the problem with any given religion is not that it is necessarily and in all cases unreasonable. It’s that it’s false.