A number of different approaches to study multivariate extremes have been developed. Arguably the most useful and flexible is the theory for the distribution of a vector variable given that one of its components is large. We build on the conditional approach of Heffernan and Tawn (2004) [13] for estimating this type of multivariate extreme property. Specifically we propose additional constraints for, and slight changes in, their model formulation. These changes in the method are aimed at overcoming complications that have been experienced with using the approach in terms of their modelling of negatively associated variables, parameter identifiability problems and drawing conditional inferences which are inconsistent with the marginal distributions. The benefits of the methods are illustrated using river flow data from two tributaries of the River Thames in the UK.