  We relate Welschinger invariants of a rational real symplectic 4-manifold
before and after a Morse simplification (i.e deletion of a sphere or a handle
of the real part of the surface). This relation is a consequence of a real
version of Abramovich-Bertram formula which computes Gromov-Witten invariants
by means of enumeration of $J$-holomorphic curves with a non-generic almost
complex structure $J$. In addition, we give some qualitative consequences of
our study, for example the vanishing of Welschinger invariants in some cases.
