There are a bunch of examples of monarchies being fundamentally involved in religion, whether as a justification for their rule ("divine right of kings") or through co-opting the religious establishment as an arm of state power (such as tsarist Russia)*.
On the other hand, there seems to be a general correlation between republicanism and secularism, two ideologies deriving from Enlightenment sources. A good example is the United States, with its explicit disestablishment clause.
I'm wondering about the feasibility of a modern democratic republic with binding ties to a religious organization--something like Isaac's eponymous Empire, but with the rather glaring difference that there wouldn't be an imperial dynasty. How well does Iran fulfill these criteria, for instance?
*I guess these are really the same thing, since the state uses its power for selfjustication. That's off-topic, though.