We develop new robust synthesis procedures for linear and open-loop exponentially stable multivariable plants subject to input nonlinearities expressed by a quadratic program and incorporating infinity-norm bounded plant uncertainties. The resulting anti-windup control falls into a class of compensators commonly termed directionality compensation. We note that the input-output maps of both the nonlinearities and the uncertainties satisfy certain integral quadratic constraints (IQCs). Thus, the anti-windup synthesis can be reduced to a feasibility problem involving a set of linear matrix inequalities (LMIs). The well-posedness condition of the algebraic loop arising from the anti-windup interconnection is equivalent to the existence and uniqueness of a solution to a convex optimization problem for which efficient solutions are well established. We demonstrate the effectiveness of the design compared to several schemes using a highly ill-conditioned benchmark example