This search, a type not previously performed at ATLAS, uses a comparison of the production cross sections for e ⁺μ ⁻ and e ⁻μ ⁺ pairs to constrain physics processes beyond the Standard Model. It uses 139fb ⁻¹ of proton–proton collision data recorded at s=13 TeV at the LHC. Targeting sources of new physics which prefer final states containing e ⁺μ ⁻ to e ⁻μ ⁺, the search contains two broad signal regions which are used to provide model-independent constraints on the ratio of cross sections at the 2% level. The search also has two special selections targeting supersymmetric models and leptoquark signatures. Observations using one of these selections are able to exclude, at 95% confidence level, singly produced smuons with masses up to 640 GeV in a model in which the only other light sparticle is a neutralino when the R-parity-violating coupling λ ₂₃₁ ^′ is close to unity. Observations using the other selection exclude scalar leptoquarks with masses below 1880 GeV when g _1R ^eu=g _1R ^μc=1, at 95% confidence level. The limit on the coupling reduces to g _1R ^eu=g _1R ^μc=0.46 for a mass of 1420 GeV.