  We prove that each semi-invariant ring of the complete triple flag of length
$n$ is an upper cluster algebra associated to an ice hive quiver. We find a
rational polyhedral cone ${\sf G}_n$ such that the generic cluster character
maps its lattice points onto a basis of the cluster algebra. As an application,
we use the cluster algebra structure to find a special minimal set of
generators for these semi-invariant rings when $n$ is small.
