Characterising the joint structure of extremes of environmental variables is important for improved understanding of those environments. Yet, many applications of multivariate extreme value analysis adopt models that assume a particular form of extremal dependence between variables without justification, or restrict attention to regions in which all variables are extreme. The conditional extremes model of Heffernan and Tawn provides one approach to avoiding these particular restrictions. Extremal marginal and dependence characteristics of environmental variables typically vary with covariates. Reliable descriptions of extreme environments should also therefore characterise any non-stationarity. A recent article by the current authors extends the conditional extremes model of Heffernan and Tawn to include covariate effects, using Fourier representations of model parameters for single periodic covariates. Here, we further extend our recent work, introducing a general purpose spline representation for model parameters as functions of multidimensional covariates, common to all inference steps. We use a non-crossing quantile regression to estimate appropriate non-stationary marginal quantiles simultaneously as functions of covariate; these are necessary as thresholds for extreme value modelling and for standardisation of marginal distributions prior to application of the conditional extremes model. Then, we perform marginal extreme value and conditional extremes modelling within a roughness-penalised likelihood framework, with cross-validation to estimate suitable model parameter roughness. Finally, we use a bootstrap re-sampling procedure, encompassing all inference steps, to quantify uncertainties in, and dependence structure of, parameter estimates and estimates of conditional extremes of one variate given large values of another. We validate the approach using simulations from known joint distributions, the extremal dependence structures of which change with covariate. We apply the approach to joint modelling of storm peak significant wave height and associated storm peak period for extra-tropical storms at a northern North Sea location, with storm direction as covariate. We evaluate the impact of incorporating directional effects on estimates for conditional return values. © 2014 John Wiley & Sons, Ltd.