Recently, it has been shown that the observed antiproton yield in heavy-ion collisions at CERN-SpS energies can be understood by multi-pionic interactions such as πππππ↔pp̅, which enforce local chemical equilibrium of the antiprotons with the nucleons and pions. Here we show that antihyperons are driven towards local chemical equilibrium with pions, nucleons and kaons on a timescale of 1-3 fm c^-1 when applying a similar argument for the antihyperons by considering the inverse channel of annihilation reactions such as Y̅ + p↔n₁π + n₂K. These multi-mesonic reactions easily explain the antihyperon yields at CERN-SpS energies as advertised in pure thermal, hadronic models without the need of a quark-gluon plasma phase. In addition, the argument also applies for AGS energies.