We examine a moduli problem for real and quaternionic vector bundles on a smooth complex projective curve with a fixed real structure, and we give a gauge-theoretic construction of moduli spaces for semistable such bundles with fixed topological type.These spaces embed onto connected subsets of real points inside a complex projective variety.We relate our point of view to previous work by Biswas et al.[BHH10], and we use this to study the Gal(C/R)-action [E] → [σ * E] on moduli varieties of stable holomorphic bundles on a complex curve with given real structure σ.We show in particular a Harnack-type theorem, bounding the number of connected components of the fixed-point set of that action by 2 g + 1, where g is the genus of the curve.In fact, taking into account all the topological invariants of σ, we give an exact count of the number of connected components, thus generalizing to rank r > 1 the results of Gross and Harris on the Picard scheme of a real algebraic curve [GH81]. Real structureson spaces of unitary connections 523 3.3.Connected components of the set of real points of the moduli scheme 529 References 533