Summary: It is shown that the Schrödinger equation can be cast in the form of two coupled real conservation equations, in Euclidean spacetime in the free case and in a five-dimensional Eisenhart geometry in the presence of an external potential. This implies a novel two-phase quantum-hydrodynamic model whose Lagrangian picture provides an exact scheme to calculate the time-dependent wavefunction from a continuum of deterministic trajectories where two points are linked by at most two trajectories. Properties of the model are examined, including the appearance of ‘entangled’ trajectories in separable states. Wavefunction constructions employing alternative two-phase models are proposed.