jats:titleAjats:scbstract</jats:sc> </jats:title>jats:pA search for the leptonic charge asymmetry (jats:inline-formulajats:alternativesjats:tex-math$$ {A}{\textrm{c}}^{\ell } $$</jats:tex-math><mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML"> mml:msubsup mml:miA</mml:mi> mml:mic</mml:mi> mml:miℓ</mml:mi> </mml:msubsup> </mml:math></jats:alternatives></jats:inline-formula>) of top-quark-antiquark pair production in association with a jats:italicW</jats:italic> boson (jats:inline-formulajats:alternativesjats:tex-math$$ t\overline{t}W $$</jats:tex-math><mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML"> mml:mit</mml:mi> mml:mover mml:mit</mml:mi> mml:mo¯</mml:mo> </mml:mover> mml:miW</mml:mi> </mml:math></jats:alternatives></jats:inline-formula>) is presented. The search is performed using final states with exactly three charged light leptons (electrons or muons) and is based on jats:inline-formulajats:alternativesjats:tex-math$$ \sqrt{s} $$</jats:tex-math><mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML"> mml:msqrt mml:mis</mml:mi> </mml:msqrt> </mml:math></jats:alternatives></jats:inline-formula> = 13 TeV proton-proton collision data collected with the ATLAS detector at the Large Hadron Collider at CERN during the years 2015–2018, corresponding to an integrated luminosity of 139 fbjats:supjats:italic−</jats:italic>1</jats:sup>. A profile-likelihood fit to the event yields in multiple regions corresponding to positive and negative differences between the pseudorapidities of the charged leptons from top-quark and top-antiquark decays is used to extract the charge asymmetry. At reconstruction level, the asymmetry is found to be −0jats:italic.</jats:italic>12 ± 0jats:italic.</jats:italic>14 (stat.) ± 0jats:italic.</jats:italic>05 (syst.). An unfolding procedure is applied to convert the result at reconstruction level into a charge-asymmetry value in a fiducial volume at particle level with the result of −0jats:italic.</jats:italic>11 ± 0jats:italic.</jats:italic>17 (stat.) ± 0jats:italic.</jats:italic>05 (syst.). The Standard Model expectations for these two observables are calculated using Monte Carlo simulations with next-to-leading-order plus parton shower precision in quantum chromodynamics and including next-to-leading-order electroweak corrections. They are jats:inline-formulajats:alternativesjats:tex-math$$ -{0.084}{-0.003}^{+0.005} $$</jats:tex-math><mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML"> mml:mo−</mml:mo> mml:msubsup mml:mn0.084</mml:mn> mml:mrow mml:mo−</mml:mo> mml:mn0.003</mml:mn> </mml:mrow> mml:mrow mml:mo+</mml:mo> mml:mn0.005</mml:mn> </mml:mrow> </mml:msubsup> </mml:math></jats:alternatives></jats:inline-formula> (scale) jats:italic±</jats:italic> 0jats:italic.</jats:italic>006 (MC stat.) and jats:inline-formulajats:alternativesjats:tex-math$$ -{0.063}_{-0.004}^{+0.007} $$</jats:tex-math><mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML"> mml:mo−</mml:mo> mml:msubsup mml:mn0.063</mml:mn> mml:mrow mml:mo−</mml:mo> mml:mn0.004</mml:mn> </mml:mrow> mml:mrow mml:mo+</mml:mo> mml:mn0.007</mml:mn> </mml:mrow> </mml:msubsup> </mml:math></jats:alternatives></jats:inline-formula> (scale) jats:italic±</jats:italic> 0jats:italic.</jats:italic>004 (MC stat.) respectively, and in agreement with the measurements.</jats:p>