  Motivated by the real version of the Gopakumar-Vafa conjecture for 3-folds,
the authors introduced in [GI] the notion of local real Gromov-Witten
invariants associated to local 3-folds over Real curves. This article is
devoted to the proof of a splitting formula for these invariants under target
degenerations. It is used in [GI] to show that the invariants give rise to a
2-dimensional Klein TQFT and to prove the local version of the real
Gopakumar-Vafa conjecture.
