  A $\mathrm{U}(p,q)$-Higgs bundle on a Riemann surface (twisted by a line
bundle) consists of a pair of holomorphic vector bundles, together with a pair
of (twisted) maps between them. Their moduli spaces depend on a real parameter
$\alpha$. In this paper we study wall crossing for the moduli spaces of
$\alpha$-polystable twisted $\mathrm{U}(p,q)$-Higgs bundles. Our main result is
that the moduli spaces are birational for a certain range of the parameter and
we deduce irreducibility results using known results on Higgs bundles. Quiver
bundles and the Hitchin-Kobayashi correspondence play an essential role.
