  A $k$-bisection of a multigraph $G$ is a partition of its vertex set into two
parts of the same cardinality such that every component of each part has at
most $k$ vertices. Cui and Liu shown that every claw-free cubic multigraph
contains a $2$-bisection, while Eom and Ozeki constructed specific
$2$-bisections with bounded number of monochromatic edges. Their bound is the
best possible for claw-free cubic simple graphs. In this note, we extend the
latter result to the larger family of claw-free cubic multigraphs
