I was going to post a snarky comment to the effect that if you discard outright religious views and cognition motivated by them, there doesn't seem to be much left to non-naturalism. But the Stanford Encyclopedia of Philosophy says it better:

There may be as much philosophical controversy about how to distinguish naturalism from non-naturalism as there is about which view is correct.

So non-naturalism looks like mostly a combination of religion and arguments over the meaning of the word "natural". As in, if we found evidence that spirits of the dead affected physical events, that would promote them to the status of natural physical phenomena. So of course everything that exists is "natural" - according to some definitions of the word.

What is there to non-naturalism that is worth the time of seriously investigating it?