In previous work, it was argued that the type IIB T⁶/₂ orientifold with a choice of flux preserving = 2 supersymmetry is dual to a class of purely geometric type IIA compactifications on abelian surface (T⁴) fibered Calabi-Yau threefolds. We provide two explicit constructions of the resulting Calabi-Yau duals. The first is a monodromy based description, analogous to F-theory encoding of Calabi-Yau geometry via 7-branes and string junctions, except for T⁴ rather than T² fibers. The second is an explicit algebro-geometric construction in which the T⁴ fibers arise as the Jacobian tori of a family of genus-2 curves. This improved description of the duality map will be a useful tool to extend our understanding of warped compactifications. We sketch applications to related work to define warped Kaluza-Klein reduction in toroidal orientifolds, and to check the modified rules for D-brane instanton zero mode counting due to the presence of flux and other D-branes. The nontrivial fundamental groups of the Calabi-Yau manifolds constructed also have potential applications to heterotic model building.