Host-virus evolutionary arms-races are driven by antagonistic interactions and often manifest as recurrent amino acid changes (i.e., positive selection) at their protein-protein interaction interfaces. Here, we investigated whether combinatorial mutagenesis of positions under positive selection in a host antiviral protein could enhance its restrictive properties. We tested ~800 variants of the human MxA protein, generated by combinatorial mutagenesis, for their ability to restrict Thogoto orthomyxovirus (THOV).We identified MxA 'super-restrictors' with increased binding to THOV NP target protein and 10-fold higher anti-THOV restriction relative to wild-type human MxA, the most potent naturally-occurring anti-THOV restrictor identified. However, MxA superrestrictors of THOV were impaired in their restriction of influenza A virus. Our findings thus reveal a breadth-versus-specificity tradeoff that constrains the adaptive landscape of antiviral proteins.