  We argue that superluminal signal propagation is possible in consistent
Poincare invariant quantum field theories in two space-time dimensions,
provided spatial parity is broken. This happens due to existence of the
``instantaneous'' causal structure, with one of the light cone variables being
a global time. In two dimensions this causal structure is invariant under the
Poincare group if one gives up the spatial parity. As a non-trivial example of
a consistent interacting quantum field theory with this causal structure we
discuss a non-linear SO(1,1) sigma-model, where SO(1,1) is the Lorentz
symmetry. We show that this theory is asymptotically free and argue that this
model is also well defined non-perturbatively, at least for some values of
parameters. It provides an example of a microscopic Poincare invariant quantum
field theory with local action, but non-local physical properties. Being
coupled to gravity this ``instantaneous'' theory mixes with the Liouville
field. If proves to be consistent, the resulting model can be used to construct
(non-critical) string theories with very unconventional properties by
introducing the instantaneous causal structure on the world-sheet.
