We identify a simple mechanism by which H-flux satisfying the modified Bianchi identity arises in garden-variety (0, 2) gauged linear sigma models. Taking suitable limits leads to effective gauged linear sigma models with Green-Schwarz anomaly cancellation. We test the quantumconsistency of a class of such effective theories by constructing an off-shell superconformal algebra, identifying unexpected topological constraints on the existence of this algebra and providing evidence that these models run to good CFTs in the deep IR when all of the constraints are satisfied.1 The term "balanced" is probably more appropriate [2], since all such 4d N = 1 compactifications come from balanced manifolds which may or may not be Kähler. The term "non-Kähler" has become standard, however, emphasizing that these manifolds need not be Kähler.2 For useful introductions and reviews of (0,2) sigma models see [10,11].5 Note that adding a GS term is not possible in a (2, 2) model. Corespondingly, the fermionic spectrum is necessarily non-chiral, which forbids any gauge anomaly. Such models necessarily have dH = 0.6 Technically, this result uses the natural Kähler structure on the toric variety. In the presence of torsion, the physical metric will in general not be this Kähler metric. They will differ, however, only by terms proportional to α ′ , the loop counting parameter in the worldsheet; for the Bianchi identity above, we need only the leading order result. Note that this argument is not reliable away from large radius -however, away from large radius, the geometric picture is itself not reliable so we should focus instead on the quantum consistency of the gauge theory.