  We give a recursive formula for purely real Welschinger invariants of real
Del Pezzo surfaces of degree $K^2\ge 3$, where in the case of surfaces of
degree $3$ with two real components we introduce a certain modification of
Welschinger invariants and enumerate exclusively the curves traced on the
non-orientable component. As an application, we prove the positivity of the
invariants under consideration and their logarithmic asymptotic equivalence, as
well as congruence modulo $4$, to genus zero Gromov-Witten invariants.
