Over the past decade, non-Hermitian, PT-symmetric Hamiltonians have been investigated as candidates for both, a fundamental, unitary, quantum theory, and open systems with a non-unitary time evolution. In this paper, we investigate the implications of the former approach in the context of the latter. Motivated by the invariance of the PT (inner) product under time evolution, we discuss the dynamics of wave-function phases in a wide range of PT-symmetric lattice models. In particular, we

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... lly show that, starting with a random initial state, a universal, gain-site location dependent locking between wave function phases at adjacent sites occurs in the PT-symmetry broken region. Our results pave the way towards understanding the physically observable implications of time-invariants in the non-unitary dynamics produced by PT-symmetric Hamiltonians.