  We show that for all $n\ge3$, an example of an $n$-partite quantum
correlation that is not genuinely multipartite nonlocal but rather exhibiting
anonymous nonlocality, that is, nonlocal but biseparable with respect to all
bipartitions, can be obtained by locally measuring the $n$-partite
Greenberger-Horne-Zeilinger (GHZ) state. This anonymity is a manifestation of
the impossibility to attribute unambiguously the underlying multipartite
nonlocality to any definite subset(s) of the parties, even though the
correlation can indeed be produced by nonlocal collaboration involving only
such subsets. An explicit biseparable decomposition of these correlations is
provided for any partitioning of the $n$ parties into two groups. Two possible
applications of these anonymous GHZ correlations in the device-independent
setting are discussed: multipartite secret sharing between any two groups of
parties and bipartite quantum key distribution that is robust against nearly
arbitrary leakage of information.
