  Let X be a compact connected Riemann surface equipped with an
anti-holomorphic involution \sigma. Let G be a connected complex reductive
affine algebraic group, and let \sigma_G be a real form of G. We consider
holomorphic principal G-bundles on X satisfying compatibility conditions with
respect to \sigma and \sigma_G. We prove that the points defined over $\mathbb
R$ of the smooth locus of a moduli space of principal G-bundles on X are
precisely these objects, under the assumption that {\rm genus}(X) > 2. Stable,
semistable and polystable bundles are defined in this context. Relationship
between any of these properties and the corresponding property of the
underlying holomorphic principal G-bundle is explored. A bijective
correspondence between unitary representations and polystable objects is
established.
