Let $(M, \omega, J)$ be a compact symplectic manifold with a compatible almost complex structure $J$, such that the symplectic form determines an integer cohomology class, ie $$ [\omega] \in H^2(M, \mathbb{Z}).$$ Does there exist an "almost homolorphic line bundle" $L \rightarrow M $ such that its first Chern class is $[\omega]$?

By "almost holomorphic line bundle", I mean a complex line bundle, whose transition data is pseudo holomorphic (ie differential commutes with J).

Note that, if M was a Kahler manifold (ie J is integrable), the answer is yes.

$\textbf{Modified Question:} $ It seems the answer to the above question is no. Here is a weaker question: Suppose $(M,\omega)$ is a compatc symplectic manifold, with integral symplectic class. Does there exist an $\omega$-compatible almost complex structure $J$, such that the answer to the above question is yes (ie there will exist an "almost homolorphic" line bunde (wrt to the J) whose first Chern class is $\omega$)?

It seems there are examples of $(M, \omega)$ such that for a generic compatible $J$ the answer is no. But still there could be some compatible $J$ for which the answer is yes.

in advance(and not assume that $J$ is integrable, for that would make $(M,\omega,J)$ be a Kähler structure, a special case that is already known), and require that the bundle $L$ be constructible by specifying local trivializations on the elements of an open cover for which the transition functions are $J$-pseudo-holomorphic. $\endgroup$ – Robert Bryant Jul 27 '13 at 19:32