  A geometrically frustrated (GF) magnet consists of localised magnetic
moments, spins, whose orientation cannot be arranged to simultaneously minimise
their interaction energies. Such materials may host novel fascinating phases of
matter, such as fluid-like states called quantum spin liquids. GF magnets have,
like all solid-state systems, randomly located impurities whose magnetic
moments may ``freeze'' at low temperatures, making the system enter a
spin-glass state. We analyse the available data for spin-glass transitions in
GF materials and find a surprising trend: the glass-transition temperature
grows with decreasing impurity concentration and reaches a finite value in the
impurity-free limit at a previously unidentified, ``hidden'', energy scale. We
propose a scenario in which the interplay of interactions and entropy leads to
a crossover in the permeability of the medium that assists glass freezing at
low temperatures. This low-temperature, ``eminuscent'', phase may obscure or
even destroy the widely-sought spin-liquid states in rather clean systems.
