In recent years, there has been a flurry of interest in gauge theoretic invariants of manifolds equipped with an involution; in particular, such invariants have been used to detect exotic RP^2-knots (Miyazawa) and to settle the non-sliceness of cables of the Figure 8 knot (Kang-Park-Taniguchi). For 3-manifolds with an involution, there is a Heegaard Floer analogue of these invariants, developed in joint work with Manolescu. However, to develop the 4-dimensional aspects of the theory, it is necessary to first show the real Heegaard Floer homology groups are natural. In the talk, we will review the key ingredients in the proof, highlighting the subtleties that arise in the equivariant setting. This is joint work in progress with C. Manolescu.