Connections on principal bundles over Kahler manifolds with antiholomorphic involution

Let M be a connected compact Kahler manifold equipped with an antiholomorphic involution tau. Let G be a complex reductive group; fix a real structure on G. We consider holomorphic principal G-bundles over M equipped with a lift of tau as an antiholomorphic involution of the total space of E-G. We extend the notion of polystability to such bundles with involution and prove that polystability is equivalent to the existence of an Einstein-Hermitian connection compatible with the involution. We also give a criterion for such a bundle over a compact Riemann surface to have a holomorphic connection compatible with the involution.