  Realizations of the holographic correspondence in String/M theory typically
involve spacetimes of the form $AdS \times Y$ where $Y$ is some internal space
which geometrizes an internal symmetry of the dual field theory, hereafter
referred to as an "$R$ symmetry". It has been speculated that areas of
Ryu-Takayanagi surfaces anchored on the boundary of a subregion of $Y$, and
smeared over the base space of the dual field theory, quantify entanglement of
internal degrees of freedom. A natural candidate for the corresponding
operators are linear combinations of operators with definite $R$ charge with
coefficients given by the "spherical harmonics'' of the internal space: this is
natural when the product spaces appear as IR geometries of higher dimensional
AdS spaces. We study clustering properties of such operators both for pure $AdS
\times Y$ and for flow geometries, where $AdS \times Y$ arises in the IR from a
different spacetime in the UV, for example higher dimensional AdS or
asymptotically flat spacetime. We show, in complete generality, that the two
point functions of such operators separated along the internal space obey
clustering properties at scales larger than the $AdS$ scale. For non-compact
$Y$, this provides a notion of approximate locality. When $Y$ is compact,
clustering happens only when the size of $Y$ is parametrically larger than the
$AdS$ scale. This latter situation is realized in flow geometries where the
product spaces arise in the IR from an asymptotically AdS geometry at UV, but
not typically when they arise near black hole horizons in asymptotically flat
spacetimes. We discuss the significance of this result for entanglement and
comment on the role of color degrees of freedom.
