  Inspired by Kronheimer and Mrowka's approach to monopole Floer homology, we
develop a model for $\mathbb{Z}/2$-equivariant symplectic Floer theory using
equivariant almost complex structures, which admits a localization map to a
twisted version of Floer cohomology in the invariant set. We then present
applications to Steenrod operations on Lagrangian Floer cohomology, and to the
Heegaard Floer homology of double covers of three-manifolds.
