Smooth real algebraic hypersurfaces of even degree in $\mathbb{RP}^4$ that are maximal (i.e. that are homologically as rich as possible in the sense of the Smith-Thom inequality) are all non-orientable.
I am trying to investigate the possible topological types and geometries of such hypersurfaces, but all the sources about 3-dimensional manifolds I browsed reduce to the orientable case and say that analogue results can be proven in the non-orientable with some adjustments. Do you know where I could find precise statements on this topic ?