Flat F-theory and friends

We discuss F-theory backgrounds associated to flat torus bundles over Ricci-flat manifolds. In this setting the F-theory background can be understood as a IIB orientifold with a large radius limit described by a supersymmetric compactification of IIB supergravity on a smooth, Ricci flat, but in general non-spin geometry. When compactified on an additional circle these backgrounds are T-dual to IIA compactifications on smooth non-orientable manifolds with a Pin-minus structure.