Belyi pair for the orientation cover of $\overline{\mathcal M_{0,5}^{\mathbb R}}$

Let $\overline{{\mathcal M}_{0,5}^{\mathbb R}}$ be the Deligne–Mumford compactification of the moduli space of genus $0$ real algebraic curves with five marked points. By ${\mathcal L}(\overline{{\mathcal M}_{0,5}^{\mathbb R}})$ we denote its orientation cover. The cell decomposition of ${\mathcal L}(\overline{{\mathcal M}_{0,5}^{\mathbb R}})$ is a dessin d'enfant of genus $4$. In this paper, we compute the Belyi pair for this dessin. In particular, it turns out that the corresponding curve is the celebrated Bring curve.