This paper presents a novel anti-windup compensator synthesis for linear parameter-varying (LPV) systems that have linear fractional transformation (LFT) representations. The LPV anti-windup synthesis is based on a right coprime factor description of the LPV plant to be controlled. It is shown that the anti-windup compensator synthesis reduces to solving a robust gain state-feedback control problem for the LPV/LFT system subject to input saturations. The synthesis takes advantage of unstructured multipliers (that is, multipliers that do not commute with the parameter structure) and exploits a detailed description of the parameter trajectory polytope. The anti-windup compensator is given in an explicit LPV/LFT form, sharing with the plant the same order and same parameter structure.