A three-parameter integrable deformation of ℤ4 permutation supercosets is constructed. These supercosets are of the form widehat{F}/{F}_0 where F0 is the bosonic diagonal subgroup of the product supergroup widehat{F}=widehat{G}times widehat{G} . They include the AdS3 × S3 and AdS3 × S3 × S3 supercosets. This deformation encompasses both the bi-Yang-Baxter deformation of the semi-symmetric space σ-model on ℤ4 permutation supercosets and the mixed flux model. Truncating the action at the bosonic level, we show that one recovers the bi-Yang-Baxter deformation of the principal chiral model plus Wess-Zumino term.