  In this paper we study the Real Gromov-Witten theory of local 3-folds over
Real curves. We show that this gives rise to a 2-dimensional Klein TQFT defined
on an extension of the category of unorientable surfaces. We use this structure
to completely solve the theory by providing a closed formula for the local RGW
invariants in terms of representation theoretic data, extending earlier results
of Bryan and Pandharipande. As a consequence we obtain the local version of the
real Gopakumar-Vafa formula that expresses the connected real Gromov-Witten
invariants in terms of integer invariants. In the case of the resolved conifold
the partition function of the RGW invariants agrees with that of the SO/Sp
Chern-Simons theory.
