  Let $\pi$ be an order-$q$-subplane of $PG(2,q^3)$ that is exterior to
$\ell_\infty$. Then the exterior splash of $\pi$ is the set of $q^2+q+1$ points
on $\ell_\infty$ that lie on an extended line of $\pi$. Exterior splashes are
projectively equivalent to scattered linear sets of rank 3, covers of the
circle geometry $CG(3,q)$, and hyper-reguli in $PG(5,q)$. In this article we
use the Bruck-Bose representation in $PG(6,q)$ to investigate the structure of
$\pi$, and the interaction between $\pi$ and its exterior splash. In $PG(6,q)$,
an exterior splash $\mathbb S$ has two sets of cover planes (which are
hyper-reguli) and we show that each set has three unique transversals lines in
the cubic extension $PG(6,q^3)$. These transversal lines are used to
characterise the carriers of $\mathbb S$, and to characterise the sublines of
$\mathbb S$.
