  We consider $\mathbb{Z}_2$-synchronization on the Euclidean lattice. Every
vertex of $\mathbb{Z}^d$ is assigned an independent symmetric random sign
$\theta_u$, and for every edge $(u,v)$ of the lattice, one observes the product
$\theta_u\theta_v$ flipped independently with probability $p$. The task is to
reconstruct products $\theta_u\theta_v$ for pairs of vertices $u$ and $v$ which
are arbitrarily far apart. Abb\'e, Massouli\'e, Montanari, Sly and Srivastava
(2018) showed that synchronization is possible if and only if $p$ is below a
critical threshold $\tilde{p}_c(d)$, and efficiently so for $p$ small enough.
We augment this synchronization setting with a model of side information
preserving the sign symmetry of $\theta$, and propose an \emph{efficient}
algorithm which synchronizes a randomly chosen pair of far away vertices on
average, up to a differently defined critical threshold $p_c(d)$. We conjecture
that $ p_c(d)=\tilde{p}_c(d)$ for all $d \ge 2$. Our strategy is to
\emph{renormalize} the synchronization model in order to reduce the effective
noise parameter, and then apply a variant of the multiscale algorithm of AMMSS.
The success of the renormalization procedure is conditional on a plausible but
unproved assumption about the regularity of the free energy of an Ising spin
glass model on $\mathbb{Z}^d$.
