They called that a millet system in the Ottoman empire

I wish the people who call for using Sharia (or, god forbid, Halakha) in the West checked out how well it works before trying it.

There is a more or less first-world country where civil law is handled by religious communities: Israel. I know that in most of the western media Israel is just an evil country to blame wherever Jews are killing Arabs, Arabs are killing Jews, Arabs are killing each other or anything else bad happens in or around the Middle East, or, in case of the more conservative media, a brave little democratic country resisting Islamic onslaught. But there is more to the place than their bad relations with Muslims, and I think that everyone who is interested in immigration and multiculturalism issues, from any perspective, should study their experience.

Anyway, one thing that they have is that the civil law is handled by the religious communities. Check out how well it worked out for them. Also check out why thousands of them get married in Cyprus every year.