Balanced colorings of networks classify robust synchrony patterns -- those that are defined by subspaces that are flow-invariant for all admissible ODEs. In symmetric networks the obvious balanced colorings are orbit colorings, where colors correspond to orbits of a subgroup of the symmetry group. All other balanced colorings are said to be exotic. We analyze balanced colorings for two closely related types of network encountered in applications: trained Wilson networks, which occur in models of binocular rivalry, and opinion networks, which occur in models of decision making. We give two examples of exotic colorings which apply to both types of network, and prove that Wilson networks with at most two learned patterns have no exotic colorings. We discuss how exotic colorings affect the existence and stability of branches for bifurcations of the corresponding model ODEs.