The equivalence between two real Seiberg-Witten Floer homologies

We show that for a real rational homology sphere $Y$ equipped with a real $\mathrm{s}\mathrm{p}\mathrm{i}{\mathrm{n}}^{\mathrm{c}}$ structure $\mathfrak{s}$, the real monopole Floer homology defined by Li and the real Seiberg-Witten Floer homology defined by Konno, Miyazawa and Taniguchi are isomorphic. As corollaries, we identify some Froyshov-type invariants and prove two Smith-type inequalities.