  The asymptotic giant branch (AGB) of the globular cluster NGC 4372 appears to
extend to unexpectedly high luminosities. We show, on the basis of proper
motions and spatial distribution, that the extended AGB is indeed a likely part
of the cluster. We also present the first spectra of the very cool (2600 K),
very luminous (8000 Lsun), very dusty, oxygen-rich, purported long-period
variable stars V1 and V2 that define the AGB tip. In particular, on the basis
of their radial velocities, we conclude that V1 and V2 are probably members. We
find that V1 and V2 are likely undergoing the superwind phase that terminates
their nuclear-burning evolution. We hypothesise that the mass-loss processes
that terminate the AGB are inhibited in NGC 4372 due to a lack of atmospheric
pulsation and the high gas-to-dust ratio in the ejecta, leading to a delay in
the associated enhanced mass loss and dust production. Previously predicted,
but never observed, this explains the high mass of the white dwarf in Pease 1
in M15 without the need to invoke a stellar merger. If commonplace, this
phenomenon has implications for the mass return from stars, the production of
carbon stars and supernovae through the Universe's history, and the AGB
contribution to light from unresolved metal-poor populations.
