  We study Ward identities and selection rules for local correlators in
disordered theories where a 0-form global symmetry of a QFT is explicitly
broken by a random coupling $h$ but it re-emerges after quenched average. We
consider $h$ space-dependent or constant. In both cases we construct the
symmetry operator implementing the group action, topological after average. In
the first case, relevant in statistical systems with random impurities, such
symmetries can be coupled to external backgrounds and can be gauged, like
ordinary symmetries in QFTs. We also determine exotic selection rules arising
when symmetries emerge after average in the IR, explaining the origin of
LogCFTs from symmetry considerations. In the second case, relevant in AdS/CFT
to describe the dual boundary theory of certain bulk gravitational theories,
the charge operator is not purely codimension-1, it can be defined only on
homologically trivial cycles and on connected spaces. Selection rules for
average correlators exist, yet such symmetries cannot be coupled to background
gauge fields in ordinary ways and cannot be gauged. When the space is
disconnected, in each connected component charge violation occurs, as expected
from Euclidean wormholes in the bulk theory. Our findings show the obstruction
to interpret symmetries emergent after average as gauged in the bulk.
