We study general two-dimensional sigma-models which do not possess manifest
Lorentz invariance. We show how demanding that Lorentz invariance is recovered
as an emergent on-shell symmetry constrains these sigma-models. The resulting
actions have an underlying group-theoretic structure and resemble Poisson-Lie
T-duality invariant actions. We consider the one-loop renormalization of these
models and show that the quantum Lorentz anomaly is absent. We calculate the
running of the couplings in general and show, with certain non-trivial
examples, that this agrees with that of the T-dual models obtained classically
from the duality invariant action. Hence, in these cases solving constraints
before and after quantization are commuting operations.Comment: V2: reference added, version to appear in Nucl. Phys.
