Belyi pair of the cell decomposition of oriented covering of the Deligne-Mumford compactification for $M^R_{0,5}$

We consider the Deligne-Mamford compactification of the moduli space of genus 0 real algebraic curves with 5 marked and numbered points and its orienting covering. The second one is a surface. So, standard cell decomposition of the original moduli space provides a dessin d'enfant on this surface. We compute the Belyi pair for this dessin. In particular, it appears that the corresponding curve is a celebrated Bring curve.