  We investigate the processes of synchronization and phase ordering in a
system of globally coupled maps possessing bistable, chaotic local dynamics.
The stability boundaries of the synchronized states are determined on the space
of parameters of the system. The collective properties of the system are
characterized by means of the persistence probability of equivalent spin
variables that define two phases, and by a magnetization-like order parameter
that measures the phase-ordering behavior. As a consequence of the global
interaction, the persistence probability saturates for all values of the
coupling parameter, in contrast to the transition observed in the temporal
behavior of the persistence in coupled maps on regular lattices. A
discontinuous transition from a non-ordered state to a collective phase-ordered
state takes place at a critical value of the coupling. On an interval of the
coupling parameter, we find three distinct realizations of the phase-ordered
state, which can be discerned by the corresponding values of the saturation
persistence. Thus, this statistical quantity can provide information about the
transient behaviors that lead to the different phase configurations in the
system. The appearance of disordered and phase-ordered states in the globally
coupled system can be understood by calculating histograms and the time
evolution of local map variables associated to the these collective states.
