Suppose M is a Hamiltonian T n -space, and that σ is an anti-symplectic involution compatible with the T action. The real locus of M is X, the fixed point set of σ. The standard example is when M is a complex manifold, σ is a complex conjugation, and X is the honest real part of M. Duistermaat uses Morse theory to give a nice description of the ordinary cohomology of X in terms of the cohomology of M. There is a residual T n 2 = (Z/2Z) n action on X, and we can use Duistermaat’s result, as well as some general facts about equivariant cohomology, to prove an equivariant analogue of Duistermaat’s theorem. In some cases, we can also extend theorems of Goresky-Kottwitz-MacPherson and Goldin-Holm to the real locus. (This is joint work with Daniel Biss and Victor Guillemin)