In this article, the extended displacement discontinuity method is extended to study the fracture problem in a three-dimensional transversely isotropic magnetoelectroelastic medium with a planar crack vertical to the plane of isotropy. Considering the electric and magnetic fields in the crack cavity, and using the Somigliana identity along with the displacement discontinuity Green’s functions, the hyper-singular boundary integral equations for the unknown displacement discontinuities across the crack face are derived. The singularity features along the crack fronts are analyzed, and the extended field intensity factors are expressed in terms of the extended displacement discontinuities on the crack face. Numerical examples on the field intensity factors are finally calculated for a vertical square crack using the exact closed-form Green’s functions due to constant displacement discontinuities over a rectangular crack element, and some interesting features are observed, which are different from the case where the crack is located in the plane of isotropy. The influence of the electric and magnetic boundary conditions along the crack face on the field intensity factors is further studied.