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Let E be a Real or Quaternionic Hermitian vector bundle over a Klein surfaceM. We study the action of the gauge group of E on the space of Galois-invariantunitary connections and we show that the closure of a semi-stable orbitcontains a unique unitary orbit of projectively flat, Galois-invariantconnections. We then use this invariant-theoretic perspective to prove aversion of the Narasimhan-Seshadri correspondence in this context:S-equivalence classes of semi-stable Real and Quaternionic vector bundes are inbijective correspondence with equivalence classes of certain appropriaterepresentations of orbifold fundamental groups of Real Seifert manifolds overthe Klein surface M.

On the Narasimhan-Seshadri correspondence for Real and Quaternionic vector bundles