quantum mechanics - How does Pauli's Exclusion Principle come into play for two (non-entangled) localized, free, non-interacting fermions approaching each other?

If two non-entangled free fermions with the same spatial wavefunctions (which do not yet overlap) throughout time approach each other, at what point during the overlapping of the two spatial wavefunctions will the two electrons have related spins (because of Pauli's Exclusion Principle), which were unrelated before the overlap?

I know, the free fermions have no quantized positions (though you can argue that the wavefunctions are superpositions of an infinite number of space "basis states", with infinitely small weight factors, while the wavefunctions grow larger through time, ending up smeared out all over space, which implies a well-defined energy), and thus no quantized energy and no energy quantum number (while their spins do have quantum numbers).

But nevertheless, I have the feeling (god forbid!) that, during the overlap of the two fermion wavefunctions (let's assume they overlap completely at a certain time), their spins must become opposite at some point (which wasn't the case before the start of the overlapping).