  We consider heterotic target space dual (0,2) GLSMs on elliptically fibered
Calabi-Yau manifolds. In this context, each half of the "dual" heterotic
theories must in turn have an F-theory dual. Moreover, the apparent
relationship between two heterotic compactifications seen in (0,2) heterotic
target space dual pairs should, in principle, induce some putative
correspondence between the dual F-theory geometries. It has previously been
conjectured in the literature that (0,2) target space duality might manifest in
F-theory as multiple K3-fibrations of the same elliptically fibered Calabi-Yau
manifold. In this work we investigate this conjecture in the context of both
6-dimensional and 4-dimensional effective theories and demonstrate that in
general, (0,2) target space duality cannot be explained by such a simple
phenomenon alone. In all cases, we provide evidence that non-geometric data in
F-theory must play at least some role in the induced F-theory correspondence,
while leaving the full determination of the putative new F-theory duality to
future work.
