In their seminal work, Xiao-Zhu have constructed exotic Hecke correspondences between the special fibers of different Shimura varieties, which do not necessarily lift to the generic fibers. As a consequence, they have constructed geometric realizations of the geometric Jacquet-Langlands correspondence, and verified generic instances of the Tate conjectures for certain Shimura varieties. Crucially, they only address Shimura varieties with good reduction, i.e., with hyperspecial level at a fixed prime. In these two talks, I will explain how to generalize their methods when allowing bad reduction (more precisely, when the level at a fixed prime is very special, which exists when the local group is quasi-split, but not necessarily unramified). In the first talk, I will explain how to construct exotic Hecke correspondences in these ramified situations between PEL type Shimura varieties. Combined with the "spherical" part of the categorical local Langlands correspondence, these yield new instances of the geometric Jacquet-Langlands correspondence. The key is to resolve the integral models by the splitting models of Pappas-Rapoport. In the second talk, I will define splitting versions of affine Deligne-Lustzig varieties, and explain how to study their irreducible components. Combining this with the aforementioned geometric Jacquet-Langlands correspondence, this can be used to verify generic instances of the Tate conjecture for certain Shimura varieties at very special level.