  We find a family of novel supersymmetric phases of the D1-D5 CFT, which in
certain ranges of charges have more entropy than all known ensembles. We also
find bulk BPS configurations that exist in the same range of parameters as
these phases, and have more entropy than a BMPV black hole; they can be thought
of as coming from a BMPV black hole shedding a "hair" condensate outside of the
horizon. The entropy of the bulk configurations is smaller than that of the CFT
phases, which indicates that some of the CFT states are lifted at strong
coupling. Neither the bulk nor the boundary phases are captured by the elliptic
genus, which makes the coincidence of the phase boundaries particularly
remarkable. Our configurations are supersymmetric, have non-Cardy-like entropy,
and are the first instance of a black hole entropy enigma with a controlled CFT
dual. Furthermore, contrary to common lore, these objects exist in a region of
parameter space (between the "cosmic censorship bound" and the "unitarity
bound") where no black holes were thought to exist.
