Is there any possibility of a Poisson Geometry version of the Fukaya category? Given a Poisson manifold Y, objects could be manifolds with isolated singularities X which have the property that TX is contained in NX maximally. The naive example would be something like the Poisson structure on R^2 which is (x^2 + y^2)(d/dx^d/dy). Branes would in this case be curves with some nodal singularity at the origin.
The morphisms could still be from holomorphic disks with respect to the standard complex structure. In principal, it seems like in this example the Fukaya category could be defined in the standard way (although maybe there is something more subtle one should do with the morphisms?). For a brane L passing through the origin there should be some interesting multiplicative structure in the algebra A(L) owing to the fact that the brane is required to remain fixed at the origin.... I would hope that the Hochschild cohomology could be related to the Poisson cohomology of the manifold though I haven't studied my example yet. What obstructions arise when trying to construct this category?