We present cosmological constraints from a joint cosmic shear analysis of the Kilo-Degree Survey (KV450) and the Dark Energy Survey (DES-Y1), which were conducted using Complete Orthogonal Sets of E / B -Integrals (COSEBIs). With COSEBIs, we isolated any B -modes that have a non-cosmic shear origin and demonstrate the robustness of our cosmological E -mode analysis as no significant B -modes were detected. We highlight how COSEBIs are fairly insensitive to the amplitude of the non-linear matter power spectrum at high k -scales, mitigating the uncertain impact of baryon feedback in our analysis. COSEBIs, therefore, allowed us to utilise additional small-scale information, improving the DES-Y1 joint constraints on S 8  =  σ 8 (Ω m /0.3) 0.5 and Ω m by 20%. By adopting a flat ΛCDM model we find S 8 = 0.755 −0.021 +0.019 , which is in 3.2 σ tension with the Planck Legacy analysis of the cosmic microwave background.