  In this paper, we study compound bi-free Poisson distributions for {\sl
two-faced families of random variables}. We prove a Poisson limit theorem for
compound bi-free Poisson distributions. Furthermore, a bi-free infinitely
divisible distribution for a two-faced family of self-adjoint random variables
can be realized as the limit of a sequence of compound bi-free Poisson
distributions of two-faced families of self-adjoint random variables. If a
compound bi-free Poisson distribution is determined by a positive number and
the distribution of a two faced family of finitely many random variables, which
has an almost sure random matrix model, and the left random variables commute
with the right random variables in the two-faced family, then we can construct
a random bi-matrix model for the compound bi-free Poisson distribution. If a
compound bi-free Poisson distribution is determined by a positive number and
the distribution of a commutative pair of random variables, we can construct an
asymptotic bi-matrix model with entries of creation and annihilation operators
for the compound bi-free Poisson distribution.
