The homotopy fixed points of Real spin bordism

We show that the 2-local splitting of spin${}^c$ bordism by Anderson–Brown–Peterson and Stong refines to a $C_2$-equivariant map in the category of spectra with $C_2$-action from Real spin bordism to a sum of (higher) connective covers of ${\mathrm{k}\mathrm{u}}_{\mathbb{R}}$ and suspensions of mod 2 Eilenberg–Mac Lane spectra. We use this to deduce a corresponding 2-local splitting of the homotopy fixed points of Real spin bordism. We also discuss prospects that arise in the genuine setting.