As abmj-jr stated, it would be highly unusual for you to be asked to show your passport in conjunction with the use of a JR pass at a station or aboard a train, although they have the right to do so at any point.

But, since foreigners are required to have their passport on their person at all times while in Japan anyway, I don't see what the "conundrum" is. Unless, of course, you have an intent to defraud.