Shepherdson's wall

The obstruction to proving the consistency of nonconstructibility by a uniform inner model construction is that, inside a universe satisfying $V=L$, every transitive inner model containing all ordinals has $L$ as its own constructible universe and therefore is the entire ambient universe. The absoluteness of constructible levels proves this by induction over successor definability and limit unions. Thus a construction valid in every model of ZF cannot always produce such an inner model satisfying $V\ne L$, and in particular cannot always produce one violating axiom of choice.