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