What foliations are symplectic foliations?

On a manifold $M$, let $\mathcal F$ be a foliation having even-dimensional orientable leaves. I was wondering under what hypothesis I can state that $\mathcal F$ is the symplectic foliation of a Poisson structure on $M$?

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One of the main goals of symplectic geometry is to answer this question in the special case where $M$ is the unique leaf... –  Tim Perutz Jan 22 '13 at 18:34
I agree with Tim that, as currently phrased, this question is an extremely hard open problem, way beyond current techniques. Perhaps though with different hypotheses, it might become a hard, open, but not completely impossible question. For example, what if the leaves are 2-dimensional? (Lefschetz fibrations play nicely with symplectic topology, after all.) What if we're allowed foliations and Poisson structures with singularities or degeneracies? (Perhaps along the lines of broken pencils on 4-manifolds?) –  Joel Fine Jan 22 '13 at 21:53
If the leaves are $2$-dimensional, then the only obstruction is whether the leaves have a consistent orientation, i.e., whether there is a nonvanishing bi-vector $\pi$ on $M$ that is tangent to the leaves of $\mathcal{F}$. In this case, $[\pi,\pi]=0$ is an identity since $\pi$ is tangent to a foliation, so any such $\pi$ will be a Poisson structure. –  Robert Bryant Jan 22 '13 at 22:38
Thanks Robert, thats very clear. (And not so hard after all!) –  Joel Fine Jan 23 '13 at 12:14