Many effects of climate change seem to be reflected not in the mean temperatures, precipitation or other environmental variables, but rather in the frequency and severity of the extreme events in the distributional tails. The most serious climate-related disasters are caused by compound events that result from an unfortunate combination of several variables. Detecting changes in size or frequency of such compound events requires a statistical methodology that efficiently uses the largest observations in the sample. We propose a simple, non-parametric test that decides whether two multivariate distributions exhibit the same tail behavior. The test is based on the entropy, namely Kullback–Leibler divergence, between exceedances over a high threshold of the two multivariate random vectors. We study the properties of the test and further explore its effectiveness for finite sample sizes. Our main application is the analysis of daily heavy rainfall times series in France (1976 -2015). Our goal in this application is to detect if multivariate extremal dependence structure in heavy rainfall change according to seasons and regions.