  We present several large classes of real Banach Lie-Poisson spaces whose
characteristic distributions are integrable, the integral manifolds being
symplectic leaves just as in finite dimensions. We also investigate when these
leaves are embedded submanifolds or when they have K\"ahler structures. Our
results apply to the real Banach Lie-Poisson spaces provided by the
self-adjoint parts of preduals of arbitrary $W^*$-algebras, as well as of
certain operator ideals.
