The modeling of multivariate extremes has received increasing recent attention because of its importance in risk assessment. In classical statistics of extremes, the joint distribution of two or more extremes has a nonparametric form, subject to moment constraints. This article develops a semiparametric model for the situation where several multivariate extremal distributions are linked through the action of a covariate on an unspecified baseline distribution, through a so-called density ratio model. Theoretical and numerical aspects of empirical likelihood inference for this model are discussed, and an application is given to pairs of extreme forest temperatures. Supplementary materials for this article are available online.