Tame and relatively elliptic ${\mathbb{C}\mathbb{P}}^1$-structures on the thrice-punctured sphere

Suppose a relatively elliptic representation $\rho$ of the fundamental group of the thrice-punctured sphere $S$ is given. We prove that all projective structures on $S$ with holonomy $\rho$ and satisfying a tameness condition at the punctures can be obtained by grafting certain circular triangles. The specific collection of triangles is determined by a natural framing of $\rho$. In the process, we show that (on a general surface $\mathrm{\Sigma }$ of negative Euler characteristics) structures satisfying these conditions can be characterized in terms of their M"obius completion, and in terms of certain meromorphic quadratic differentials.