  We show that for $X$ a proper $\mathrm{CAT}(-1)$ space there is a maximal
open subset of the horofunction compactification of $X\times X$ with respect to
the maximum metric that compactifies the diagonal action of an infinite
quasi-convex group of the isometries of $X$. We also consider the product
action of two quasi-convex representations of an infinite hyperbolic group on
the product of two different proper $\mathrm{CAT}(-1)$ spaces.
