Triangulation of the map of a $G$-manifold to its orbit space

Let $G$ be a Lie group and $M$ a smooth proper $G$-manifold. Let $pi:Mto M/G$ denote the natural map to the orbit space. Then there exist a PL manifold $P$, a polyhedron $L$ and homeomorphisms $tau:Pto M$ and $\sigma:M/Gto L$ such that $\sigma\circpi\circ\tau$ is PL. If $M$ and the $G$-action are of analytic class, we can choose subanalytic $\tau$ and then unique $P$ and $L$.