Abstract: 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 C. 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, I will review the construction of our invariants, and discuss a sketch of the proof of naturality, highlighting the subtleties that arise in the equivariant setting. This is joint work in progress with Manolescu.