  We extend previous relativistic mean-field (RMF) calculations of multi-$\bar
K$ nuclei, using vector boson fields with SU(3) PPV coupling constants and
scalar boson fields constrained phenomenologically. For a given core nucleus,
the resulting $\bar K$ separation energy $B_{\bar K}$, as well as the
associated nuclear and $\bar K$-meson densities, saturate with the number
$\kappa$ of $\bar K$ mesons for $\kappa > \kappa_{\rm sat} \sim 10$. Saturation
appears robust against a wide range of variations, including the RMF nuclear
model used and the type of boson fields mediating the strong interactions.
Because $B_{\bar K}$ generally does not exceed 200 MeV, it is argued that
multi-$\bar K$ nuclei do not compete with multihyperonic nuclei in providing
the ground state of strange hadronic configurations and that kaon condensation
is unlikely to occur in strong-interaction self-bound strange hadronic matter.
Last, we explore possibly self-bound strange systems made of neutrons and
${\bar K}^0$ mesons, or protons and $K^-$ mesons, and study their properties.
