  We study the structure of the Liouville quantum gravity (LQG) surfaces that
are cut out as one explores a conformal loop-ensemble CLE$_{\kappa'}$ for
$\kappa'$ in $(4,8)$ that is drawn on an independent $\gamma$-LQG surface for
$\gamma^2=16/\kappa'$. The results are similar in flavor to the ones from our
paper dealing with CLE$_{\kappa}$ for $\kappa$ in $(8/3,4)$, where the loops of
the CLE are disjoint and simple. In particular, we encode the combined
structure of the LQG surface and the CLE$_{\kappa'}$ in terms of stable
growth-fragmentation trees or their variants, which also appear in the
asymptotic study of peeling processes on decorated planar maps.
  This has consequences for questions that do a priori not involve LQG
surfaces: Our previous paper "CLE percolations" described the law of interfaces
obtained when coloring the loops of a CLE$_{\kappa'}$ independently into two
colors with respective probabilities $p$ and $1-p$. This description was
complete up to one missing parameter $\rho$. The results of the present paper
about CLE on LQG allow us to determine its value in terms of $p$ and $\kappa'$.
It shows in particular that CLE$_{\kappa'}$ and CLE$_{16/\kappa'}$ are related
via a continuum analog of the Edwards-Sokal coupling between FK$_q$ percolation
and the $q$-state Potts model (which makes sense even for non-integer $q$
between $1$ and $4$) if and only if $q=4\cos^2(4\pi /\kappa')$. This provides
further evidence for the long-standing belief that CLE$_{\kappa'}$ and
CLE$_{16/\kappa'}$ represent the scaling limits of FK$_q$ percolation and the
$q$-Potts model when $q$ and $\kappa'$ are related in this way. Another
consequence of the formula for $\rho(p,\kappa')$ is the value of half-plane arm
exponents for such divide-and-color models (a.k.a. fuzzy Potts models) that
turn out to take a somewhat different form than the usual critical exponents
for two-dimensional models.
