  This is the second paper of a series dedicated to the study of Poisson
structures of compact types (PMCTs). In this paper, we focus on regular PMCTs,
exhibiting a rich transverse geometry. We show that their leaf spaces are
integral affine orbifolds. We prove that the cohomology class of the leafwise
symplectic form varies linearly and that there is a distinguished polynomial
function describing the leafwise sympletic volume. The leaf space of a PMCT
carries a natural Duistermaat-Heckman measure and a Weyl type integration
formula holds. We introduce the notion of a symplectic gerbe, and we show that
they obstruct realizing PMCTs as the base of a symplectic complete isotropic
fibration (a.k.a. a non-commutative integrable system).
