An asymmetric Schottky tunneling source field-effect transistor (STS FET) is a prospective device structure to suppress the short-channel effect. Recently, we succeeded in the fabrication and operation of a Ge-STS n-channel FET with TiN and PtGe asymmetric metal source/drain (S/D) on a bulk Ge substrate. However, the Ge-STS p-channel FET has not been demonstrated yet. In this study, we fabricated an asymmetric metal S/D FET with the same S/D structure on a bulk Ge and a Ge-on-Insulator (GOI) substrate. The GOI was made by using the Smart-CutTM technique. The device fabricated on a bulk Ge did not operate. On the other hand, the fabricated FET on a GOI, which has a taper-shaped TiN/Ge source interface, showed STS p-FET behavior. These results suggest that the carrier injection can be improved by the optimization of the device structure. As an auxiliary effect, conventional metal-oxide-semiconductor (MOS) FET operation was also observed, thanks to GOI introduction. We demonstrated both STS mode and MOSFET mode operation in the same device on GOI.