  The Welschinger invariants of real rational algebraic surfaces are natural
analogues of the Gromov-Witten invariants, and they estimate from below the
number of real rational curves passing through prescribed configurations of
points. We establish a tropical formula for the Welschinger invariants of four
toric Del Pezzo surfaces, equipped with a non-standard real structure. Such a
formula for real toric Del Pezzo surfaces with a standard real structure (i.e.,
naturally compatible with the toric structure) was established by Mikhalkin and
the author. As a consequence we prove that, for any real ample divisor $D$ on a
surfaces $\Sigma$ under consideration, through any generic configuration of
$c_1(\Sigma)D-1$ generic real points there passes a real rational curve
belonging to the linear system $|D|$.
