We present silicon-compatible trigated p-Ge/i-Si/n-Si axial heteronanowire tunneling field-effect transistors (TFETs), where on-state tunneling occurs in the Ge drain section, while off-state leakage is dominated by the Si junction in the source. Our TFETs have high I(ON) ~ 2 μA/μm, fully suppressed ambipolarity, and a subthreshold slope SS ~ 140 mV/decade over 4 decades of current with lowest SS ~ 50 mV/decade. Device operation in the tunneling mode is confirmed by three-dimensional TCAD simulation. Interestingly, in addition to the TFET mode, our devices work as standard nanowire FETs with a good I(ON)/I(OFF) ratio when the source-drain junction is forward-biased. The improved transport in both biasing modes confirms the benefits of utilizing bandgap engineered axial nanowires for enhancing device performance.