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We consider thermodynamic and dynamic phase transitions in plaquette spin models of glasses. The thermodynamic transitions involve coupled (annealed) replicas of the model. We map these coupled-replica systems to a single replica in a magnetic field, which allows us to analyze the resulting phase transitions in detail. For the triangular plaquette model (TPM), we find for the coupled-replica system a phase transition between high- and low-overlap phases, occurring at a coupling ε∗(T), which vanishes in the low-temperature limit. Using computational path sampling techniques, we show that a single TPM also displays "space-time" transitions between active and inactive dynamical phases. These first-order dynamical transitions occur at a critical counting field sc(T) 0 that appears to vanish at zero temperature in a manner reminiscent of the thermodynamic overlap transition. In order to extend the ideas to three dimensions, we introduce the square pyramid model, which also displays both overlap and activity transitions. We discuss a possible common origin of these various phase transitions, based on long-lived (metastable) glassy states.