@NickL Yes, I would be interested in seeing such a proof, especially if one can work out the associated polytopes in a systematic way.

It is not clear what topological or geometrical features are relevant to the problem I'm looking at. At this step, I am looking for analogues of the classical decomposition theorems, and maybe a survey of the specific properties of the non-orientable manifolds from the main families of examples we know. Also, I read that only five out of the eight geometries could be realized by non-orientable manifolds, but I have not been able to find the origine and the proof of this statement.

Thank you everyone for your comments !

Thank you very much for taking the time to write such a detailed answer. I was trying to work with unions with the hope of obtaining families of toric hypersurfaces degenerating to a union of toric divisors, but I see now why I rather consider Minkowski sums.

Thank you for this very nice answer.