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Physically, a wormhole spacetime would have to be a solution of the EFE, and the only solution of the EFE with zero curvature everywhere is flat Minkowski spacetime. So any physical spacetime with wormholes present would have to have nonzero curvature.

Are you sure about that? I would think that a cylindrical (or toroidal) universe would be consistent with the EFE. After all, as far as the field equations are concerned, there is no difference between a toroidal universe and one whose mass/energy density just happens to be periodic in space.