Mustafin degenerations of six points in the projective plane and self-duality

Mustafin varieties provide a framework for degenerating the projective plane together with marked points, governed by the combinatorics of Bruhat-Tits buildings. Such limit objects are parametrized by the boundary points of a projective compactification of the moduli space of $n$ points in $\mathbb{P}^2$ constructed by Schaffler-Tevelev. In this paper, we study the locus of this compactification generically parametrizing self-dual configurations, or equivalently, six points lying on a conic. We prove that this locus is a tropical compactification of the moduli space of self-dual configurations, with associated fan a refinement of the self-dual tropical Grassmannian introduced by Geiger-Hashimoto-Sturmfels-Vlad. We show that this compactification is a blow up of $\overline{\mathrm{M}}_{0,6}$ and we classify the objects it parametrizes: degenerations of $\mathbb{P}^2$ equipped with a stable $6$-pointed rational curve that records the limiting conic.