This search, a type not previously performed at ATLAS, uses a comparison of the production cross sections for e(+)mu(-) and e(-)mu(+) pairs to constrain physics processes beyond the Standard Model. It uses 139 fb(-1) of proton-proton collision data recorded at root s = 13 TeV at the LHC. Targeting sources of new physics which prefer final states containing e(+)mu(-) and e(-)mu(+), 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 lambda(23)(1)' is close to unity. Observations using the other selection exclude scalar leptoquarks with masses below 1880 GeV when g(1R)(eu) = g(1R)(mu c) = 1, at 95% confidence level. The limit on the coupling reduces to g(1R)(eu) = g(1R)(mu c) = 0.46 for a mass of 1420 GeV.