You must first establish that there are any real phenomena that science can not discuss, and that religion can.

"The Lord, thy God, is a just God."

"Do unto others as you would have others do unto you."

"The Kingdom of Heaven is within you."

"Look you unto the lilies of the field. They neither toil, neither do they strain. Yet verily, Solomon in all his glory was not arrayed like one of these."

Just a few off the top of my head, apologies for any misquotes.

Of course you inserted the word "real" in there which is a complete weasel, because of course, religion and science can go to the barricades about real issues like evolution versus young-Earth creationism, so th e non-overlapping part does not have to apply, really; but it is my opinion they SHOULD be non-overlapping. People get uncomfortable with non-coherent or contradictory data, though. If a person holds simultaneously in his mind the belief that the principles of Darwinism are correct AND that the species were created on the nth day by the hand of an Almighty, does he suffer from dissonance? Or does he find some higher plane from which to view the contradictions comfortably? Or does he just subdivide, holding belief (a) on Monday through Friday, belief (b) on Sunday, and on Saturday concentrate on sports?

A