  Given a Hamiltonian system $ (M,\omega, G,\mu) $ where $(M,\omega)$ is a
symplectic manifold, $G$ is a compact connected Lie group acting on
$(M,\omega)$ with moment map $ \mu:M \rightarrow\mathfrak{g}^{*}$, then one may
construct the symplectic quotient $(M//G, \omega_{red})$ where $M//G :=
\mu^{-1}(0)/G$. Kirwan used the norm-square of the moment map, $|\mu|^2$, as a
G-equivariant Morse function on $M$ to derive formulas for the rational Betti
numbers of $M//G$.
  A real Hamiltonian system $(M,\omega, G,\mu, \sigma, \phi) $ is a Hamiltonian
system along with a pair of involutions $(\sigma:M \rightarrow M, \phi:G
\rightarrow G) $ satisfying certain compatibility conditions. These imply that
the fixed point set $M^{\sigma}$ is a Lagrangian submanifold of $(M,\omega)$
and that $M^{\sigma}//G^{\phi} := (\mu^{-1}(0) \cap M^{\sigma})/G^{\phi}$ is a
Lagrangian submanifold of $(M//G, \omega_{red})$. In this paper we prove
analogues of Kirwan's Theorems that can be used to calculate the
$\mathbb{Z}_2$-Betti numbers of $M^{\sigma}//G^{\phi} $. In particular, we
prove (under appropriate hypotheses) that $|\mu|^2$ restricts to a
$G^{\phi}$-equivariantly perfect Morse-Kirwan function on $M^{\sigma}$ over
$\mathbb{Z}_2$ coefficients, describe its critical set using explicit real
Hamiltonian subsystems, prove equivariant formality for $G^{\phi}$ acting on
$M^{\sigma}$, and combine these results to produce formulas for the
$\mathbb{Z}_2$-Betti numbers of $M^{\sigma}//G^{\phi}$.
