  We present a theoretical analysis of spatial correlations in a
one-dimensional driven-dissipative non-equilibrium condensate. Starting from a
stochastic generalized Gross-Pitaevskii equation, we derive a noisy
Kuramoto-Sivashinsky equation for the phase dynamics. For sufficiently strong
interactions, the coherence decays exponentially in close analogy to the
equilibrium Bose gas. When interactions are small on a scale set by the
nonequilibrium condition, we find through numerical simulations a crossover
between a Gaussian and exponential decay with peculiar scaling of the coherence
length on the fluid density and noise strength.
