  This article presents and analyses an exponential integrator for the
stochastic Manakov equation, a system arising in the study of pulse propagation
in randomly birefringent optical fibers. We first prove that the strong order
of the numerical approximation is $1/2$ if the nonlinear term in the system is
globally Lipschitz-continuous. Then, we use this fact to prove that the
exponential integrator has convergence order $1/2$ in probability and almost
sure order $1/2$, in the case of the cubic nonlinear coupling which is relevant
in optical fibers. Finally, we present several numerical experiments in order
to support our theoretical findings and to illustrate the efficiency of the
exponential integrator as well as a modified version of it.
