Abstract: The electroweak production and subsequent decay of single top quarks is determined by the properties of the Wtb vertex. This vertex can be described by the complex parameters of an effective Lagrangian. An analysis of angular distributions of the decay products of single top quarks produced in the t -channel constrains these parameters simultaneously. The analysis described in this paper uses 4.6 fb−1 of proton-proton collision data at (Formula presented.) TeV collected with the ATLAS detector at the LHC. Two parameters are measured simultaneously in this analysis. The fraction f1 of decays containing transversely polarised W bosons is measured to be 0.37 ± 0.07 (stat.⊕syst.). The phase δ− between amplitudes for transversely and longitudinally polarised W bosons recoiling against left-handed b-quarks is measured to be −0.014π ± 0.036π (stat.⊕syst.). The correlation in the measurement of these parameters is 0.15. These values result in two-dimensional limits at the 95% confidence level on the ratio of the complex coupling parameters gR and VL, yielding Re[gR/VL] ∈ [−0.36, 0.10] and Im[gR/VL] ∈ [−0.17, 0.23] with a correlation of 0.11. The results are in good agreement with the predictions of the Standard Model.[Figure not available: see fulltext.]