We apply recent results on robust global asymptotic stabilization of the attitude of a single rigid body to the problem of globally synchronizing the attitude of a network of rigid bodies using a decentralized strategy. The proposed hybrid feedback scheme relies on the communication of a binary logic variable between each pair of neighboring rigid bodies that determines the orientation of a torque component acting to reduce their relative error. Through a hysteretic switch of this logic variable, the hybrid feedback achieves global synchronization under the assumption that the network is connected and acyclic. The hysteresis eliminates chattering while preventing the “unwinding phenomenon” apparent in some quaternion-based attitude control schemes. The results are exercised in a numerical example.