This article deals with the general theory of games played over uncontrolled event trees, i.e., games where the transition from one node to another is nature's decision and cannot be influenced by the players' actions. The solution concept for this class of games was introduced under the name of S-adapted equilibrium where S stands for sample of realizations of the random process. In this paper, it is assumed that the players also face a coupled constraint at each node, and therefore the relevant solution concept is the normalized equilibrium à la Rosen. Existence and uniqueness conditions for this equilibrium are provided, as well as a stochastic-control formulation of the game and a maximum principle. A simple illustrative example in environmental economics is presented.