  The Desale-Ramanan Theorem is an isomorphism between the moduli space of rank
two vector bundles over complex hyperelliptic curve and the variety of linear
subspaces in an intersection of two quadrics. We prove a real version of this
theorem for the moduli space of real vector bundles over a real hyperelliptic
curve. We then apply this result to study the topology of the moduli space,
proving that it is relatively spin and identifying the diffeomorphism type for
genus two curves. Our results lay the groundwork for future study of the
quantum homology of these moduli spaces.
