We demonstrate tunable bistability and a strong negative differential resistance in InAs/GaSb core-shell nanowire devices embedding a radial broken-gap heterojunction. Nanostructures have been grown using a catalyst-free synthesis on a Si substrate. Current-voltage characteristics display a top peak-to-valley ratio of 4.8 at 4.2 K and 2.2 at room temperature. The Esaki effect can be modulated-or even completely quenched-by field effect, by controlling the band bending profile along the azimuthal angle of the radial heterostructure. Hysteretic behavior is also observed in the presence of a suitable resistive load. Our results indicate that high-quality broken-gap devices can be obtained using Au-free growth.