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 139 fb-1of proton–proton collision data recorded at vs=13TeV 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 640GeV in a model in which the only other light sparticle is a neutralino when the Rparity-violating coupling ?231is close to unity. Observations using the other selection exclude scalar leptoquarks with masses below 1880GeV when geu 1R =gµc 1R =1, at 95% confidence level. The limit on the coupling reduces to geu 1R =gµc 1R =0.46for a mass of 1420GeV.