Coupled complex Monge-Amp\`ere equations on Fano horosymmetric manifolds

We give necessary and sufficient conditions for existence of solutions to a general system of complex Monge-Amp`ere equations on Fano horosymmetric manifolds. In particular, we get necessary and sufficient conditions for existence of coupled K"ahler-Ricci solitons, Mabuchi metrics and twisted K"ahler-Einstein metrics in terms of combinatorial data of the manifold.