  We explore topological transitions in the type of propagation of surface
electromagnetic modes in massive anisotropic tilted Dirac systems. The presence
of tilting and mass gives rise to an indirect band gap that strongly modifies
the joint density of states compared to the gapless system. New van Hove
singularities appear, and the interplay between intra and interband transitions
leads to an anisotropic optical conductivity with imaginary parts acquiring
opposite signs in orthogonal directions, opening the possibility of having
hyperbolic propagation of plasmons. Isofrequency contours and low plasmon
losses, as obtained from the dispersion relation, show that transitions between
purely anisotropic quasi-elliptical and well-defined, highly directional,
hyperbolic modes are attainable only when tilt and mass coexist. Fine-tuning in
the THz regime suggests that this might be useful as a dynamical probe of
concurring tilt and mass in Dirac materials.
