The Pauli principle isn't a force at all. It's a symmetry, or at least a special case of one. Also, it has nothing to do what causes magnets to repel. If you violated the symmetry, magnets would still work.
I'm not sure the distinction between a force and a symmetry is a useful one. Any use of "force" in modeling physics can be equivalently expressed via conservation of linear momentum, which itself is equivalent to the fact that the laws of physics are symmetric in translation (i.e. translation-invariant, i.e. have the same form when the origin of the coordinate system everything is expressed in is moved around).
Literally, for any force, you can say, "that's just the playing out of a necessary symmetry in the laws of physics".
I thought this video was a really good question dissolving by Richard Feynman. But it's in 240p! Nobody likes watching 240p videos. So I transcribed it. (Edit: That was in jest. The real reasons are because I thought I could get more exposure this way, and because a lot of people appreciate transcripts. Also, Paul Graham speculates that the written word is universally superior than the spoken word for the purpose of ideas.) I was going to post it as a rationality quote, but the transcript was sufficiently long that I think it warrants a discussion post instead.
Here you go: