  We demonstrate that superconducting aluminium nano-bridges can be driven into
a state with complete suppression of the critical supercurrent via
electrostatic gating. Probing both in- and out-of-plane magnetic field
responses in the presence of electrostatic gating can unveil the mechanisms
that primarily cause the superconducting electric field effects. Remarkably, we
find that a magnetic field, independently of its orientation, has only a weak
influence on the critical electric field that identifies the transition from
the superconducting state to a phase with vanishing critical supercurrent. This
observation points to the absence of a direct coupling between the electric
field and the amplitude of the superconducting order parameter or $2\pi$-phase
slips via vortex generation. The magnetic field effect observed in the presence
of electrostatic gating is described within a microscopic model where a
spatially uniform inter-band $\pi$-phase is stabilized by the electric field.
Such an intrinsic superconducting phase rearrangement can account for the
suppression of the supercurrent, as well as for the weak dependence of the
critical magnetic fields on the electric field.
