The reaction $\overline{p}p\ensuremath{\rightarrow}\overline{\ensuremath{\Lambda}}\ensuremath{\Lambda}\ensuremath{\rightarrow}\overline{p}{\ensuremath{\pi}}^{+}p{\ensuremath{\pi}}^{\ensuremath{-}}$ has been measured with high statistics at a beam momentum of ${p}_{\overline{p}}=1.637\mathrm{GeV}/c$. The use of a transversely polarized frozen-spin target combined with the self-analyzing property of $\ensuremath{\Lambda}/\overline{\ensuremath{\Lambda}}$ decay allows access to unprecedented information on the spin structure of the interaction. The most general spin-scattering matrix can be written in terms of 11 real parameters for each bin of scattering angle; each of these parameters is determined with reasonable precision. From these results, all conceivable spin correlations are determined with inherent self-consistency. Good agreement is found with the few previously existing measurements of spin observables in $\overline{p}p\ensuremath{\rightarrow}\overline{\ensuremath{\Lambda}}\ensuremath{\Lambda}$ near this energy. Existing theoretical models do not give good predictions for those spin observables that had not been previously measured.