Allowable complex saddles with real charges in the gravitational index

We present a new supersymmetric, non-extremal complexification of asymptotically $\mathrm{A}\mathrm{d}\mathrm{S}$ black holes, characterized by real charges and multiple complex horizons. We investigate whether these solutions define admissible supersymmetric saddles of the gravitational path integral by applying the Kontsevich-Segal-Witten (KSW) criterion, and compare the resulting gravitational constraints with the convergence conditions of the dual superconformal index. For most of the cases studied, the KSW criterion in the asymptotic region agrees with the index convergence, while the full bulk condition is more stringent and excludes saddles whose chemical potentials lie within the index-convergence domain.