>I'd be shocked if this could produce correlated values. But if you can think of some way to test this, be my guest.

That is what I'm thinking on!  We *must* still get correlations between our RNGs at some level with this scheme, because by chance there will always be times where we happen to re-use the same parts of the underlying RNG sequence in each of our RVs.  The question is whether or not these correlations are enough to hurt us, and I think a test is needed.  As I said, this is for another issue...