An idea that I believe I have posted somewhere before: a theodemocracy, where the priesthoods of the country’s various religions vote to select the leadership. The size of a priest-elector contingency depends on how big the religion is in the nation. How much control parishioners get over who their priest-electors are depends on the faith. The irreligious and unrecognized religions (usually newer belief systems and those too small to count) are unrepresented. As for the nation itself, I’m thinking either India or maybe even a remnant of Alexander’s empire, which turned Buddhist with Alexander but has believers from Greco-Roman pagans to Hindus.
Isn't Lebanon organized along those lines? Where the President is Christian, and the Prime Minister is Muslim?