  We develop a generalized TAP approach for the multi-species version of the
spherical mixed $p$-spin models. In particular, we prove a generalized TAP
representation for the free energy at any overlap vector which is
multi-samplable in an appropriate sense. Moreover, we show that if a
multi-samplable overlap is maximal, then the TAP correction is equal to an
analogue of the well-known Onsager reaction term. Finally, in a companion paper
we use the results from the current paper to compute the free energy at any
temperature for all multi-species pure $p$-spin models, assuming the free
energy converges.
