  We prove that if a compact K\"ahler Poisson manifold has a symplectic leaf
with finite fundamental group, then after passing to a finite \'etale cover, it
decomposes as the product of the universal cover of the leaf and some other
Poisson manifold. As a step in the proof, we establish a special case of
Beauville's conjecture on the structure of compact K\"ahler manifolds with
split tangent bundle.
