Computation of the first Stiefel-Whitney class for the variety $\overline{{\mathcal{M}}_{0,n}^{\mathbb{R}}}$

We compute the class which is Poincare dual to the first Stiefel-Whitney class for the Deligne-Mumford compactification of the moduli space of real algebraic curves of genus 0 with n marked and numbered points in terms of the natural cell decomposition of the variety under consideration.