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 5 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 the genus 4. In this paper we compute the Belyi pair for this dessin. In particular, it appears that the corresponding curve is a celebrated Bring curve.