We present a class of hyperbolic systems modeling two-phase, two-velocity flows that can be obtained from Hamilton's principle of stationary action, under conventional constraints: conservation of the mass for each component. The governing equations imply the conservation of total momentum and total energy. The hyperbolicity is guaranteed for small relative velocities, i.e. close to equilibrium, by the presence of turbulence through a Reynolds stress tensor. This is the main novelty of the paper. The incorporation of this tensor in the equations is made by the addition of a turbulent term in the potential energy that appears in the Lagrangian of the model. Various forms of the turbulent term are proposed, and the ones leading to a hyperbolic system of equations are characterized by a general criterion. The presence of the Reynolds stress tensor is crucial for the hyperbolicity. Indeed, if it is removed from the Lagrangian, the resulting equations are not hyperbolic for small relative velocities.