  We study topological strings on non-commutative resolutions of singular
Calabi-Yau threefolds that are double covers of $\mathbb{P}^3$, ramified over
determinantal octic surfaces. Using conifold transitions to complete
intersections in toric ambient spaces, we prove that any small resolution has
2-torsional exceptional curves and is necessarily non-K\"ahler. The same
transitions imply that M-theory develops a $\mathbb{Z}_2$ gauge symmetry on the
singular space. We then construct gauged linear sigma models with hybrid phases
that flow to the worldsheet theories of strings propagating on the
determinantal double solids in the presence of a flat but topologically
non-trivial B-field. Localizing the sphere partition function allows us to
calculate the fundamental periods of the mirror Calabi-Yau manifolds, then we
check agreement with the periods of the Borisov-Li mirrors. We find that the
corresponding variations of Hodge structure either correspond to one of the 14
hypergeometric cases or to a double cover thereof. We then use mirror symmetry
and integrate the holomorphic anomaly equations to calculate
$\mathbb{Z}_2$-refined Gopakumar-Vafa invariants for several examples.
