Real spectral triples over noncommutative Bieberbach manifolds

We classify and construct all real spectral triples over noncommutative Bieberbach manifolds, which are restrictions of irreducible, real, equivariant spectral triples over the noncommutative three-torus. We show that, in the classical case, the constructed geometries correspond exactly to spin structures over Bieberbach manifolds and the Dirac operators constructed for a flat metric.