Morita equivalence of Nijenhuis structures

We introduce Morita equivalence for Nijenhuis groupoids and for their infinitesimal counterparts, establishing a global-to-infinitesimal correspondence under the Lie functor. A special case is that of holomorphic Lie groupoids and algebroids. We use our framework to enhance the known Morita equivalences for quasi-symplectic groupoids and Dirac structures with compatible Nijenhuis structures. Finally, subject to certain conditions, we prove that the modular class of Poisson-Nijenhuis manifolds is invariant under Morita equivalence.