  We consider measurements of exclusive rare semi-tauonic $b$-hadron decays,
mediated by the $b \rightarrow s \tau^+ \tau^-$ transition, at a future high
energy circular electron-positron collider (FCC-$ee$). We argue that the high
boosts of $b$-hadrons originating from on-shell $Z$ boson decays allow for a
full reconstruction of the decay kinematics in hadronic $\tau$ decay modes (up
to discrete ambiguities). This, together with the potentially large statistics
of $Z\to b\bar b$, opens the door for the experimental determination of $\tau$
polarizations in these rare $b$-hadron decays. In light of the current
experimental situation on lepton flavor universality in rare semileptonic $B$
decays, we discuss the complementary short-distance physics information carried
by the $\tau$ polarizations and suggest suitable theoretically clean
observables in the form of single- and double-$\tau$ polarization asymmetries.
