  Recent works have shown that random triangulations decorated by critical
($p=1/2$) Bernoulli site percolation converge in the scaling limit to a
$\sqrt{8/3}$-Liouville quantum gravity (LQG) surface (equivalently, a Brownian
surface) decorated by SLE$_6$ in two different ways:
  1. The triangulation, viewed as a curve-decorated metric measure space
equipped with its graph distance, the counting measure on vertices, and a
single percolation interface converges with respect to a version of the
Gromov-Hausdorff topology.
  2. There is a bijective encoding of the site-percolated triangulation by
means of a two-dimensional random walk, and this walk converges to the
correlated two-dimensional Brownian motion which encodes SLE$_6$-decorated
$\sqrt{8/3}$-LQG via the mating-of-trees theorem of Duplantier-Miller-Sheffield
(2014); this is sometimes called $\textit{peanosphere convergence}$.
  We prove that one in fact has $\textit{joint}$ convergence in both of these
two senses simultaneously. We also improve the metric convergence result by
showing that the map decorated by the full collection of percolation interfaces
(rather than just a single interface) converges to $\sqrt{8/3}$-LQG decorated
by CLE$_6$ in the metric space sense.
  This is the first work to prove simultaneous convergence of any random planar
map model in the metric and peanosphere senses. Moreover, this work is an
important step in an ongoing program to prove that random triangulations
embedded into $\mathbb C$ via the so-called $\textit{Cardy embedding}$ converge
to $\sqrt{8/3}$-LQG.
