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If the wavefunctions of two non-interacting fermions overlap completely for a while after which they separate can it be that these overlapping pieces become entangled, just like their spin functions will become entangled, no matter how far they are apart (transcending spacetime)? This implies that measuring the fermion positions (when their wavefunctions are far away from each other) of these once overlapping wavefunctions will show a correlation between the positions of the fermions (just like the spins show a correlation), so these positions also transcend spacetime. So can it be that once overlapping wavefunctions will be correlated or entangled after separation?
How can this entanglement show itself? Well, maybe the distance between the measured values (when far apart) of the position on the fermions stays constant in time, after their mean distance has been taken into account (or instead of being constant these measured distances vary in a certain defined way because their wavefunctions spread out).

Which means that if we measure the position of one fermion (while the corresponding wavefunctions are very far apart from each other), the position of the other will have a certain position, or interval of positions, just like measuring the spin of one of them gives the opposite spin of the other fermion.

Or will the positions of the fermions each just obey the probabilities of the separate wavefunctions, which is the standard view?

Have experiments been done to look for this or is it ruled out in the first place?

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No, two overlapping non-interactive fields won't become entangled. To become entangled they would need to interact. The reason is, that to form an entangled state from the tensor product of two separable states, one needs a global unitary operation that operates on the entire tensor product and can thus produce correlation among the degrees of freedom of the two states. The evolution of two non-interactive fermions is described by local unitary operations that operate separately on the two fields and thus would not produce entanglement.

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