We present cosmological constraints from a joint cosmic shear analysis of the Kilo-Degree Survey (KV450) and the Dark Energy Survey (DES-Y1), conducted using Complete Orthogonal Sets of E/B-Integrals (COSEBIs). With COSEBIs we isolate any B-modes which have a non-cosmic shear origin and demonstrate the robustness of our cosmological E-mode analysis as no significant B-modes are 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, allow us to utilise additional small-scale information, improving the DES-Y1 joint constraints on S 8 = σ8(Ωm/0:3)0:5 and Ωm by 20%. Adopting a flat CDM model we find S 8 = 0:755+0:019-0:021, which is in 3:2σ tension with the Planck Legacy analysis of the cosmic microwave background.