  We study stochastic Poisson integrators for a class of stochastic Poisson
systems driven by Stratonovich noise. Such geometric integrators preserve
Casimir functions and the Poisson map property. For this purpose, we propose
explicit stochastic Poisson integrators based on a splitting strategy, and
analyse their qualitative and quantitative properties: preservation of Casimir
functions, existence of almost sure or moment bounds, asymptotic preserving
property, and strong and weak rates of convergence. The construction of the
schemes and the theoretical results are illustrated through extensive numerical
experiments for three examples of stochastic Lie--Poisson systems, namely:
stochastically perturbed Maxwell--Bloch, rigid body and sine--Euler equations.
