We explicitly realize an internal action of the symplectic cactus group, originally defined by Halacheva for any finite dimensional complex reductive Lie algebra, on crystals of Kashiwara-Nakashima tableaux. Our methods include a symplectic version of jeu de taquin due to Sheats and Lecouvey, symplectic reversal, and virtualization due to Baker. As an application, we define and study a symplectic version of the Berenstein-Kirillov group and show that it is a quotient of the symplectic cactus group.