  In this paper, we enumerate with signs the holomorphic maps between Real
Riemann surfaces. We relate the signs and the numbers obtained to the Real
Gromov-Witten theory of the target. For a finite group $G$ with a non-trivial
morphism to $\{ \pm 1 \}$, we present a signed count of principal $G$-bundles
yielding a Klein Topological Quantum Field Theory.
