  We continue the investigation of the isomorphism problem for multiplier
algebras of reproducing kernel Hilbert spaces with the complete Nevanlinna-Pick
property. In contrast to previous work in this area, we do not study these
spaces by identifying them with restrictions of a universal space, namely the
Drury-Arveson space. Instead, we work directly with the Hilbert spaces and
their reproducing kernels. In particular, we show that two multiplier algebras
of Nevanlinna-Pick spaces on the same set are equal if and only if the Hilbert
spaces are equal. Most of the article is devoted to the study of a special
class of Nevanlinna-Pick spaces on homogeneous varieties. We provide a complete
answer to the question of when two multiplier algebras of spaces of this type
are algebraically or isometrically isomorphic. This generalizes results of
Davidson, Ramsey, Shalit, and the author.
