An adaptive approach is proposed to address the problem of robustification (and performance loss reduction) of anti-windup (AW) compensation in the presence of large uncertainties. The approach relies on the augmentation of the AW compensator with an adaptive robustifying filter; a constructive procedure is provided for designing such a filter. The use of adaptation allows to reduce the conservativeness of previous robust approaches to what is needed to preserve stability with respect to the uncertainty actually present on the plant during operation, as detected by input and output plant measurement.