A regular Poisson manifold can be described as a foliated space supporting a tangentially symplectic form. Examples of foliations will be presented in the talk that are not induced by any Poisson structure although the basic obstructions vanish. On the other hand, tangentially symplectic structures are solutions of a certain partial differential relation that can be shown to satisfy the $h$-principle when some natural condition on the underlying foliated space is fulfilled. The proof of this statement can be sketched if time permits.