By a non-orientable Riemann surface ${\cal C}$, I mean a compact non-orientable two-manifold without boundary that is endowed with a conformal structure.

Such objects have a moduli space that is somewhat like the more familiar moduli space of oriented Riemann surfaces, but this moduli space is itself non-orientable. (See corollary 2.3 of https://arxiv.org/abs/1309.0383, where this is proved with a simple explicit example.)

My question is this: Suppose that ${\cal C}$ is endowed with a $pin^+$ structure. (This is one of the two possible analogs of a spin structure in the non-orientable case, the other being a $pin^-$ structure.) I've come to suspect that the moduli space of non-orientable Riemann surfaces with a $pin^+$ structure is itself orientable. I wonder if this is a known result and where it might be found.