  Let $(\Sigma,\tau)$ denote a Riemann surface of genus $g \geq 2$ equipped
with an anti-holomorphic involution $\tau$. In this paper we study the topology
of the moduli space $M(r,\xi)^\tau$ of stable Real vector bundles over
$(\Sigma,\tau)$ of rank $r$ and fixed determinant $\xi$ of degree coprime to
$r$.
  We prove that $M(r,\xi)^{\tau}$ is an orientable and monotone Lagrangian
submanifold of the complex moduli space $M(r,\xi)$ so it determines an object
in the appropriate Fukaya category. We derive recursive formulas for the mod
$2$ Betti numbers of $M(r,\xi)^\tau$ and compute mod $p$ Betti numbers for odd
$p$ through a range of degrees. We deduce that if $r$ is even and $ g >>0$,
then $M(r,\xi)^{\tau}$ and $M(r,\xi')^{\tau}$ have non-isomorphic cohomology
groups unless $\xi$ and $\xi'$ have equivalent Stieffel-Whitney classes modulo
automorphisms of $(\Sigma,\tau)$. If $r$ is even, and $g>>0$ is even, we prove
that the Betti numbers of $M(r,\xi)^{\tau}$ distinguish topological types of
$(\Sigma, \tau; \xi)$. If $r=2$ and $g$ is odd, we compute all mod $p$ Betti
numbers of $M(2,\xi)^\tau$.
  MR 32L05, 14P25.
