  The assumption that quantum systems relax to a stationary state in the
long-time limit underpins statistical physics and much of our intuitive
understanding of scientific phenomena. For isolated systems this follows from
the eigenstate thermalization hypothesis. When an environment is present the
expectation is that all of phase space is explored, eventually leading to
stationarity. Notable exceptions are decoherence-free subspaces that have
important implications for quantum technologies and have so far only been
studied for systems with a few degrees of freedom. Here we identify simple and
generic conditions for dissipation to prevent a quantum many-body system from
ever reaching a stationary state. We go beyond dissipative quantum state
engineering approaches towards controllable long-time non-stationarity
typically associated with macroscopic complex systems. This coherent and
oscillatory evolution constitutes a dissipative version of a quantum
time-crystal. We discuss the possibility of engineering such complex dynamics
with fermionic ultracold atoms in optical lattices.
