  We present a new approach to local independence in spin glasses, i.e. the
phenomenon that any fixed subset of coordinates is asymptotically independent
in the thermodynamic limit. The approach generalizes the rigorous cavity method
from Talagrand by considering multiple cavity sites. Under replica-symmetric
conditions of thin-shell and overlap concentration, the cavity fields are
revealed to be asymptotically independent, conditionally on the disorder, which
in turn leads to local independence. Conversely, it is shown that local
independence implies those replica-symmetric properties. The framework is
general enough to encompass the classical and soft spin ($[-1,1]$)
Sherrington-Kirkpatrick models, as well as the Gardner spin glasses.
