A bstract Measurements of the inclusive and normalised differential cross sections are presented for the production of single top quarks in association with a W boson in proton-proton collisions at a centre-of-mass energy of 13 TeV. The data used were recorded with the CMS detector at the LHC during 2016–2018, and correspond to an integrated luminosity of 138 fb − 1 . Events containing one electron and one muon in the final state are analysed. For the inclusive measurement, a multivariate discriminant, exploiting the kinematic properties of the events is used to separate the signal from the dominant $$ \textrm{t}\overline{\textrm{t}} $$ <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML"> <mml:mi>t</mml:mi> <mml:mover> <mml:mi>t</mml:mi> <mml:mo>¯</mml:mo> </mml:mover> </mml:math> background. A cross section of $$ 79.2\pm 0.9{\left(\textrm{stat}\right)}_{-8.0}^{+7.7}\left(\textrm{syst}\right)\pm 1.2\left(\textrm{lumi}\right) $$ <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML"> <mml:mn>79.2</mml:mn> <mml:mo>±</mml:mo> <mml:mn>0.9</mml:mn> <mml:msubsup> <mml:mfenced> <mml:mtext>stat</mml:mtext> </mml:mfenced> <mml:mrow> <mml:mo>−</mml:mo> <mml:mn>8.0</mml:mn> </mml:mrow> <mml:mrow> <mml:mo>+</mml:mo> <mml:mn>7.7</mml:mn> </mml:mrow> </mml:msubsup> <mml:mfenced> <mml:mtext>syst</mml:mtext> </mml:mfenced> <mml:mo>±</mml:mo> <mml:mn>1.2</mml:mn> <mml:mfenced> <mml:mtext>lumi</mml:mtext> </mml:mfenced> </mml:math> pb is obtained, consistent with the predictions of the standard model. For the differential measurements, a fiducial region is defined according to the detector acceptance, and the requirement of exactly one jet coming from the fragmentation of a bottom quark. The resulting distributions are unfolded to particle level and agree with the predictions at next-to-leading order in perturbative quantum chromodynamics.