  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.
