  We show that if a multigraph $G$ with maximum edge-multiplicity of at most
$\frac{\sqrt{n}}{\log^2 n}$, is edge-coloured by $n$ colours such that each
colour class is a disjoint union of cliques with at least $2n + o(n)$ vertices,
then it has a full rainbow matching, that is, a matching where each colour
appears exactly once. This asymptotically solves a question raised by Clemens,
Ehrenm\"uller and Pokrovskiy, and is related to problems on algebras of sets
studied by Grinblat in [Grinblat 2002].
