  Evert and Helton proved that real free spectrahedra are the matrix convex
hulls of their absolute extreme points. However, this result does not extend to
complex free spectrahedra, and we examine multiple ways in which the analogous
result can fail. We also develop some local techniques to determine when matrix
convex sets are not (duals of) free spectrahedra, as part of a continued study
of minimal and maximal matrix convex sets and operator systems. These results
apply to both the real and complex cases.
