  The Zero-Range Process, in which particles hop between sites on a lattice
under conserving dynamics, is a prototypical model for studying real-space
condensation. Within this model the system is critical only at the transition
point. Here we consider a non-conserving Zero-Range Process which is shown to
exhibit generic critical phases which exist in a range of creation and
annihilation parameters. The model also exhibits phases characterised by
mesocondensates each of which contains a subextensive number of particles. A
detailed phase diagram, delineating the various phases, is derived.
