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

We compute the class $W_{n-4}(\overline{\mathcal M_{0,n}^\mathbb R})$, which is Poincaré dual to the first Stiefel–Whitney class for the variety $\overline{\mathcal M_{0,n}^\mathbb R}$ in terms of the natural cell decomposition of $\overline{\mathcal M_{0,n}^\mathbb R}$.