The design of reliable path-following controllers is a key ingredient for successful deployment of self-driving vehicles. This controller-design problem is especially challenging for a general 2-trailer with a car-like tractor due to the vehicles structurally unstable joint-angle kinematics in backward motion and the car-like tractors curvature limitations which can cause the vehicle segments to fold and enter a jackknife state. Furthermore, advanced sensors with a limited field of view have been proposed to solve the joint-angle estimation problem online, which introduce additional restrictions on which vehicle states that can be reliably estimated. To incorporate these restrictions at the level of control, a model predictive path-following controller is proposed. By taking the vehicles physical and sensing limitations into account, it is shown in real-world experiments that the performance of the proposed path-following controller in terms of suppressing disturbances and recovering from non-trivial initial states is significantly improved compared to a previously proposed solution where the constraints have been neglected.