  We study the matrix range of a tuple of compact operators on a Hilbert space
and examine the notions of minimal, nonsingular, and fully compressed tuples.
In this pursuit, we refine previous results by characterizing nonsingular
compact tuples in terms of matrix extreme points of the matrix range. Further,
we find that a compact tuple $A$ is fully compressed if and only if it is
multiplicity-free and the Shilov ideal is trivial, which occurs if and only if
$A$ is minimal and nonsingular. Fully compressed compact tuples are therefore
uniquely determined up to unitary equivalence by their matrix ranges. We also
produce a proof of this fact which does not depend on the concept of
nonsingularity.
