I can't think of a situation where two parties would want to generate the same sequence of random numbers without communication, however, if I ever need it, this seems like a really interesting solution.
Could you give an example? The zero knowledge proofs I know are of the type: here's an X, tell me f(secret, X) - which doesn't require any shared information (apart from the secret you're verifying).
What I was thinking was that the random number from GPS (or some sequence derived from it) could serve as the challenge, basically akin to a 2FA scheme like HOTP. Or a variety of other schemes, hinging on the trait that you can't predict the next output of the random number from GPS.