Economically responsible mitigation of multivariate extreme risks-such as extreme rainfall over large areas, large simultaneous variations in many stock prices, or widespread breakdowns in transportation systems-requires assessing the resilience of the systems under plausible stress scenarios. This paper uses Extreme Value Theory (EVT) to develop a new approach to simulating such multivariate extreme events. Specifically, we assume that after transformation to a standard scale the distribution of the random phenomenon of interest is multivariate regular varying and use this to provide a sampling procedure for extremes on the original scale. Our procedure combines a Wasserstein-Aitchison Generative Adversarial Network (WA-GAN) to simulate the tail dependence structure on the standard scale with joint modeling of the univariate marginal tails on the original scale. The WA-GAN procedure relies on the angular measure-encoding the distribution on the unit simplex of the angles of extreme observations-after transformation to Aitchison coordinates, which allows the Wasserstein-GAN algorithm to be run in a linear space. Our method is applied both to simulated data under various tail dependence scenarios and to a financial data set from the Kenneth French Data Library. The proposed algorithm demonstrates strong performance compared to existing alternatives in the literature, both in capturing tail dependence structures and in generating accurate new extreme observations.