  Given a probability measure on a finitely generated group, its Martin
boundary is a way to compactify the group using the Green function of the
corresponding random walk. It is known from the work of W. Woess that when a
finitely supported random walk on a free product of abelian groups is adapted
to the free product structure, the Martin boundary coincides with the geometric
boundary. The main goal of this paper is to deal with non-adapted finitely
supported random walks, for which there is no explicit formula for the Green
function. Nevertheless, we show that the Martin boundary still coincides with
the geometric boundary. We also prove that the Martin boundary is minimal.
