  We construct a class of stable SU(5) bundles on an elliptically fibered
Calabi-Yau threefold with two sections, a variant of the ordinary Weierstrass
fibration, which admits a free involution. The bundles are invariant under the
involution, solve the topological constraint imposed by the heterotic anomaly
equation and give three generations of Standard Model fermions after symmetry
breaking by Wilson lines of the intermediate SU(5) GUT-group to the Standard
Model gauge group. Among the solutions we find some which can be perturbed to
solutions of the Strominger system. Thus these solutions provide a step toward
the construction of phenomenologically realistic heterotic flux
compactifications via non-Kahler deformations of Calabi-Yau geometries with
bundles. This particular class of solutions involves a rank two hidden sector
bundle and does not require background fivebranes for anomaly cancellation.
