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. 



Does there always exist a line bundle whose Chern class represents an integer symplectic form?
The answer is already 'no' in dimension $4$. The generic almost complex structure compatible with a symplectic structure in dimension $4$ does not admit any pseudoholomorphic functions (in your sense) other than the constants, so, for such data $(M,\omega,J)$, you are asking whether there is a line bundle $L$ that can be described by constant transition functions such that its first Chern class is $[\omega]$. 


To see that this is not always possible, consider the special case in which $M$ is simply-connected, and choose $J$ generically compatible with $\omega$ such that the sheaf of $J$-pseudoholomorphic functions is the constant sheaf. Then any $L$ that can be built with constant transition functions will inherit from that construction a flat connection $\nabla$ with holonomy in $\mathbb{C}^\ast$. Since $M$ is simply connected and nabla is flat, the holonomy will be trivial, so $L$ will have a nonvanishing parallel section and hence be the trivial line bundle, so its first Chern class will be zero.


