We propose a method to control networks involving both cooperative and competitive interactions simultaneously under proximity and collision-avoidance constraints. The control design is of gradient type, using a barrier-Lyapunov function. Then, under the assumption that the network is undirected and structurally balanced, we establish asymptotic stability of the leaderless and leader-follower bipartite formation-consensus manifold. We assume that the agents are modeled by secondorder integrators, but we also demonstrate the utility of our theoretical findings via numerical simulations on a problem of formation-consensus control of nonholonomic vehicles.