Welschinger invariants of toric Del Pezzo surfaces with non-standard real structures

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 $|D|$.