We introduce the notion of weakly log-canonical Poisson structures on positive varieties with potentials. Such a Poisson structure is log-canonical up to terms dominated by the potential. To a compatible real form of a weakly logcanonical Poisson variety, we assign an integrable system on the product of a certain real convex polyhedral cone (the tropicalization of the variety) and a compact torus. We apply this theory to the dual Poisson-Lie group G* of a simply-connected semisimple complex Lie group G.
