  Generalizing Culler and Vogtmann's Outer Space for the free group, Guirardel
and Levitt construct an Outer Space for a free product of groups. We completely
characterize when this space (or really its simplicial spine) supports an
equivariant piecewise-Euclidean or piecewise-hyperbolic CAT(0) metric. Our
results are mostly negative, extending thesis work of Bridson and related to
thesis work of Cunningham. In particular, provided the dimension of the spine
is at least three, it is never CAT(0). Surprisingly, we exhibit one family of
free products for which the Outer Space is two-dimensional and \emph{does}
support an equivariant CAT(0) metric.
