= Solution
The necessary and sufficient condition is the <transverse-wall cocompactness criterion>: there must be finitely many $G$-orbits of finite <transverse wall collections>. Equivalently, their cardinalities must be uniformly bounded and, for every cardinality, there must be only finitely many orbits.
Indeed, an $n$-cube of the <dual cube complex of a wallspace> is dual to an $n$-element collection of pairwise crossing walls, and this correspondence respects the $G$-action and passage to faces. If there are finitely many orbits of transverse collections, there are finitely many cube orbits, so the quotient $G\backslash C$ is a finite cube complex and is compact. Conversely, if the action is cocompact, a compact fundamental set meets only finitely many open unit cubes. Hence there are finitely many cube orbits and therefore finitely many orbits of their dual transverse wall collections.
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