Let $L$ be a lattice of signature $(1,n)$. Suppose I have a (probably infinite index) subgroup $\Gamma\subset O^+(L)$ of the isometries of $L$ which preserve the positive cone $\mathcal{C}^+\subset L\otimes \mathbb{R}$. Now, suppose that there is an open set $A\subset \mathbb{P}\mathcal{C}^+\cong \mathbb{H}^n$ of hyperbolic space defined by an intersection of (possibly infinitely many) half-spaces such that the following hold:

1) $\Gamma\cdot A=A$

2) There is a finite set of polyhedra $P_i\subset A$ (which may contain up to one cusp of $\mathbb{H}^n$) such that $P_i\rightarrow \Gamma\backslash A$ is injective on the interior of $P_i$ and the images of $P_i$ cover all of $\Gamma\backslash A$.

My question is, is there necessarily a polyhedral fundamental domain for the action of $\Gamma$ on $A$? I know this is a rather specific question, but of course if there is a more general result in this direction, I would love to know. For the purposes of this question, a polyhedron is a subset of $\mathbb{P}(L\otimes \mathbb{R})$ defined by a finite set of linear inequalities. For instance, $\mathbb{P}\mathcal{C}^+$ is NOT a polyhedron.