  We study lattices acting on $\mathrm{CAT}(0)$ spaces via their commensurated
subgroups. To do this we introduce the notions of a graph of lattices and a
complex of lattices giving graph and complex of group splittings of
$\mathrm{CAT}(0)$ lattices. Using this framework we characterise irreducible
uniform $(\mathrm{Isom}(\mathbb{E}^n)\times T)$-lattices by $C^\ast$-simplicity
and give a necessary condition for lattices in products with a Euclidean factor
to be biautomatic. We also construct non-residually finite uniform lattices
acting on arbitrary products of right-angled buildings and non-biautomatic
lattices acting on the product of $\mathbb{E}^n$ and a right-angled building.
