  We explicitly realize an internal action of the symplectic cactus group,
recently defined by Halacheva for any complex, reductive, finite-dimensional
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. In addition two relations for
symplectic Berenstein-Kirillov group are given that do not follow from the
defining relations of the symplectic cactus group.
