  A split system on a multiset $\mathcal M$ is a set of bipartitions of
$\mathcal M$. Such a split system $\mathfrak S$ is compatible if it can be
represented by a tree in such a way that the vertices of the tree are labelled
by the elements in $\mathcal M$, the removal of each edge in the tree yields a
bipartition in $\mathfrak S$ by taking the labels of the two resulting
components, and every bipartition in $\mathfrak S$ can be obtained from the
tree in this way. In this contribution, we present a novel characterization for
compatible split systems, and for split systems admitting a unique tree
representation. In addition, we show that a conjecture on compatibility stated
in 2008 holds for some large classes of split systems.
