In the early 80s, Kirwan proved a relationship between the equivariant cohomology of a Hamiltonian action on a symplectic manifold, and the cohomology of its symplectic quotient. I present an version of this relationship for symplectic manifolds equipped with an anti-symplectic involution, relating the cohomology of corresponding fixed point Lagrangian submanifolds. I then apply this result to study the topology of moduli spaces of vector bundles and Higgs bundles over a real algebraic curve, in the style of Atiyah-Bott and Hitchin.