Quantum oscillations of response functions in high magnetic fields tend to reveal some of the most interesting properties of metals. In particular, the oscillation phase shift is sensitive to topological band features, thereby helping to identify the presence of Weyl fermions. In this work, we predict a characteristic parameter dependence of the phase shift for Weyl fermions with tilted and overtilted dispersion (type-I and type-II Weyl fermions) and an arbitrary topological charge (multi-Weyl fermions). For type-II Weyl fermions our calculations capture the case of magnetic breakthrough between the electron and the hole part of the dispersion. Here, the phase shift turns out to depend only on the quantized topological charge due to the cancellation of nonuniversal contributions from the electron and the hole part.
