  The local behavior of typical solutions of random constraint satisfaction
problems (CSP) describes many important phenomena including clustering
thresholds, decay of correlations, and the behavior of message passing
algorithms. When the constraint density is low, studying the planted model is a
powerful technique for determining this local behavior which in many examples
has a simple Markovian structure. Work of Coja-Oghlan, Kapetanopoulos,
M\"{u}ller (2020) showed that for a wide class of models, this description
applies up to the so-called condensation threshold.
  Understanding the local behavior after the condensation threshold is more
complex due to long-range correlations. In this work, we revisit the random
regular NAE-SAT model in the condensation regime and determine the local weak
limit which describes a random solution around a typical variable. This limit
exhibits a complicated non-Markovian structure arising from the space of
solutions being dominated by a small number of large clusters. This is the
first description of the local weak limit in the condensation regime for any
sparse random CSPs in the one-step replica symmetry breaking (1RSB) class. Our
result is non-asymptotic, and characterizes the tight fluctuation $O(n^{-1/2})$
around the limit. Our proof is based on coupling the local neighborhoods of an
infinite spin system, which encodes the structure of the clusters, to a
broadcast model on trees whose channel is given by the 1RSB belief-propagation
fixed point. We believe that our proof technique has broad applicability to
random CSPs in the 1RSB class.
