  Any multiplicative quiver variety is endowed with a Poisson structure
constructed by Van den Bergh through reduction from a Hamiltonian quasi-Poisson
structure. The smooth locus carries a corresponding symplectic form defined by
Yamakama through quasi-Hamiltonian reduction. In this note, we include the
Poisson structure as part of a pencil of compatible Poisson structures on the
multiplicative quiver variety. The pencil is defined by reduction from a pencil
of Hamiltonian quasi-Poisson structures which has dimension $\ell(\ell-1)/2$,
where $\ell$ is the number of arrows in the underlying quiver. For each element
of the pencil, we exhibit the corresponding compatible symplectic or
quasi-Hamiltonian structure. We comment on analogous constructions for
character varieties and quiver varieties. This formalism is applied to the spin
Ruijsenaars-Schneider phase space in order to explain the compatibility of two
Poisson structures that have recently appeared in the literature.
