Vibrational energy harvesters, and more particularly piezoelectric devices, usually feature two limitations: their narrow frequency band and their relatively limited conversion abilities. The first issue is typically addressed using mechanical nonlinearities introducing a polynomial stiffness, while the second concern may be overcome through nonlinear electrical interfaces. However, combining these two approaches is not straightforward because of the backward coupling which can make the energy harvesting process enhancement affecting the nonlinear mechanical behavior. Hence, the purpose of this paper is to expose and deeply investigate, both theoretically and experimentally, an energy harvesting system coupling such mechanical and electrical nonlinearities. Such a multinonlinear device is achieved through the combination of tristability on the mechanical side and a technique of synchronous discharge on the electrical aspect. Analytical investigations together with experimental measurements therefore show that the energy extraction magnification brought by the electrical interface may compromise the bandwidth enhancement offered by the nonlinear stiffness due to the coupled multiphysic nature of the device that yields damping effect. Ways of controlling the trade-off between bandwidth and power conversion are also devised through the introduction of a phase delay in the electrical switch command.