Analysing the extremes of multi-dimensional data is a difficult task for many reasons, e.g. the wide range of extremal dependence structures and the scarcity of the data. Some popular approaches that account for various extremal dependence types are based on asymptotically motivated models so that there is a probabilistic underpinning basis for extrapolating beyond observed levels. Among these efforts, Heffernan and Tawn developed a methodology for modelling the distribution of a d-dimensional variable when at least one of its components is extreme. Their approach is based on a series (i=1,…,d) of conditional distributions, in which the distribution of the rest of the vector is modelled given that the ith component is large. This model captures a wide range of dependence structures and is applicable to cases of large d. However their model suffers from a lack of self-consistency between these conditional distributions and so does not uniquely determine probabilities when more than one component is large. This paper looks at these unsolved issues and makes proposals which aim to improve the efficiency of the Heffernan–Tawn model in practice. Tests based on simulated and financial data suggest that the proposed estimation method increases the self-consistency and reduces the RMSE of the estimated coefficient of tail dependence.