  In this paper we study a natural generalization of symplectic toric manifolds
in the context of regular Poisson manifolds of compact types. To be more
precise, we consider a class of multiplicity-free Hamiltonian actions by
regular proper symplectic groupoids that we call faithful. Given such a
groupoid, we classify its faithful multiplicity-free Hamiltonian actions in
terms of what we call Delzant subspaces of its orbit space -- certain
`suborbifolds with corners' satisfying the Delzant condition relative to the
integral affine orbifold structure of the orbit space. This encompasses both
the classification of symplectic toric manifolds (due to Delzant) in terms of
Delzant polytopes and the classification of proper Lagrangian fibrations over
an integral affine base manifold (due to Duistermaat) in terms of a sheaf
cohomology group. Each Delzant subspace comes with an orbifold version of this
cohomology, the degree one part of which classifies faithful multiplicity-free
Hamiltonian actions with momentum map image equal to the Delzant subspace,
provided there exists such an action. The obstruction to existence is encoded
by a degree two class in this cohomology: the Lagrangian Dixmier-Douady class.
In addition to the above, we introduce another invariant, which leads to a
variation of our classification result involving only classical sheaf
cohomology and the group cohomology of certain modules for the isotropy groups
of the groupoid.
