  Let $(X,c_X)$ be a convex projective surface equipped with a real structure.
The space of stable maps $\bar{\mathcal{M}}_{0,k}(X,d)$ carries different real
structures induced by $c_X$ and any order two element $\tau$ of permutation
group $S_k$ acting on marked points. Each corresponding real part
$\R_{\tau}\bar{\mathcal{M}}_{0,k}(X,d)$ is a real normal projective variety. As
the singular locus is of codimension bigger than two, these spaces thus carry a
first Stiefel-Whitney class for which we determine a representative in the case
$k=c_1(X)d-1$ where $c_1(X)$ is the first Chern class of $X$. Namely, we give a
homological description of these classes in term of the real part of boundary
divisors of the space of stable maps.
