  Time-reversal symmetry enables shaping input waves to control output waves in
many linear and nonlinear systems; however energy dissipation violates such
symmetry. We consider a saturated multimode fiber amplifier in which light
generates heat flow and suffers nonlinear thermo-optical scattering, breaking
time-reversal symmetry. We identify a spacetime symmetry which maps the target
output back to an input field. This mapping employs phase conjugation, gain and
absorption substitution but not time reversal, and holds in steady-state and
for slowly varying inputs. Our results open the possibility of output control
of a saturated multimode fiber amplifier.
