This paper presents a nonlinear path tracking controller for autonomous bi-steerable (four-wheel steering or 4WS) vehicles, allowing high precision tracking even when the reference path proposes fast varying curvature. Indeed, such paths are very common as soon as vehicles have to avoid obstacles in cluttered environments. Considering the well-known bicycle model, the sole reference path usually defined for the rear wheel is replaced by two synchronized paths, introducing a new way to calculate the expected yaw rate of the vehicle without numerical derivatives. Equations describing the motion of the vehicle with respect to this "dual-path" are presented and used to design the proposed control law. Then, simulations and experiments with the electrical public transport vehicle EZ10 demonstrate the controller ability to precisely follow complex trajectories.