  We investigate existence and uniqueness for the stochastic liquid crystal
flow driven by colored noise on the two-dimensional torus. After giving a
natural uniqueness criterion, we prove local solvability in $L^p$-based spaces,
for every $p>2.$ Thanks to a bootstrap principle together with a
Gy\"ongy-Krylov-type compactness argument, this will ultimately lead us to
prove the existence of a particular class of global solutions which are
partially regular, strong in the probabilistic sense, and taking values in the
"critical space" $L^2\times H^1.$
