  Commability is the finest equivalence relation between locally compact groups
such that $G$ and $H$ are equivalent whenever there is a continuous proper
homomorphism $G \to H$ with cocompact image. Answering a question of Cornulier,
we show that all non-elementary locally compact groups acting geometrically on
locally finite simplicial trees are commable, thereby strengthening previous
forms of quasi-isometric rigidity for trees. We further show that 6
homomorphisms always suffice, and provide the first example of a pair of
locally compact groups which are commable but without commation consisting of
less than 6 homomorphisms. Our strong quasi-isometric rigidity also applies to
products of symmetric spaces and Euclidean buildings, possibly with some
factors being trees.
