For AdS/CFT to make sense, symmetries must match between the AdS side and the CFT side. Gauge symmetries are redundancies, not symmetries, therefore the CFT can have a (large) gauge symmetry, say $SU(N)$, that isn't visible on the gravity side.
But unless I'm mistaken, only the small gauge transformations correspond to redundancies. Global $SU(N)$ should still be a "true" symmetry of the theory, and therefore exist on both sides of the duality. Is this correct?
If yes, how can I see that global $SU(N)$ is a symmetry of the bulk gravity/string theory?