The fundamental technique of symplectic topology is the theory of pseudo-holomorphic curves. One studies maps $u$ from a Riemann surface into a symplectic manifold, equipped with an almost complex structure tamed by the symplectic form, such that $Du$ is complex-linear. Numerous algebraic structures can be built from such maps: Gromov-Witten invariants, Hamiltonian Floer cohomology, Floer cohomology of pairs of Lagrangian submanifolds, and most elaborate of all, $A_\infty$-structures on Lagrangian Floer cochains (Fukaya categories).
Though the basic theory of pseudo-holomorphic curves makes sense on more general almost-complex manifolds, the presence of the symplectic structure is vital for Gromov compactness to be applicable. Without this, your curves are liable to vanish into thin air. None of the algebraic structures I mentioned have been developed on almost complex manifolds, nor on Poisson manifolds. So far as I know, there's no theory of global Poisson topology in the sense that there is global symplectic topology, and it's not clear that there can be such a theory.
There are situations where one might reasonably hope to find relations between Poisson geometry and symplectic topology, but in those situations it may be wise to go via intermediate constructions. For instance, a version of the derived Fukaya category of $T^{\ast} L$ was shown by Nadler to be equivalent to the derived category of constructible sheaves on $L$, and I'm told that that category is related to deformation quantization of $T^{\ast} L$ - something which truly does belong to Poisson geometry.