Suppose $p:(M,J)\rightarrow (N,I)$ is a submersion between smooth manifolds M and N such that:

$(M,J)$ is an almost-complex manifold.

$(N,I)$ is a complex manifold where $I$ is the integrable almost-complex structure.

- $p$ is almost-holomorphic, that is $dp\circ J=I\circ dp.$
- The fibers of $p,$ which are $J$-holomorphic by the previous condition, are holomorphic in the usual sense. That is, the almost complex structure $J$ restricted to any fiber $p^{-1}(x)$ for $x\in M$ is integrable.

Is $J$ integrable?

I believe that the answer to this question is likely to be no, though I am having trouble cooking up a counter-example. The only immediate consequence I see is that the Nijenhuis tensor on $M$ must take values in vertical vector fields. I am primarily interested in the case where $M$ and $N$ are not compact so standard techniques of deformation theory of compact complex manifold don't easily apply, though I don't know the answer in the case when $p$ is proper either. My google searches haven't turned up anything useful yet, so I'd be very interested to hear about any additional conditions one could add to this list, namely additional structure on $M,$ which guarantees that $J$ is integrable if, as I suspect, the answer to my original question is negative. Thanks in advance for any comments.