This paper studies Nash equilibria in two player continuous time stochastic differential games with diffusion control, and where the Brownian motions driving the state processes are correlated. We consider zero-sum ranking games, in the sense that the criteria to optimize only depends on the difference of the two players' state processes. We explicitly compute the players' equilibrium strategies, depending on the correlation of the Brownian motions driving the two state equations: in particular, if the correlation coefficient is smaller than some explicit threshold, then the equilibrium strategies consist of strong controls, whereas if the correlation exceeds the threshold, then the optimal controls are mixed strategies. To do so, we rely on a relaxed formulation of the game based on solutions to martingale problems, allowing the players to randomize their actions.