Coupled K\"ahler-Ricci solitons on toric Fano manifolds

We prove a necessary and sufficient condition in terms of the barycenters of a collection of polytopes for existence of coupled K"ahler-Einstein metrics on toric Fano manifolds. This confirms the toric case of a coupled version of the Yau-Tian-Donaldson conjecture. We also obtain a necessary and sufficient condition for existence of torus-invariant solutions to a system of soliton type equations on toric Fano manifolds. Some of these solutions provide natural candidates for the large time limits of a certain geometric flow generalizing the K"ahler-Ricci flow.