  We construct a class of solvable models for 2+1D quantum critical points by
attaching 1+1D conformal field theories (CFTs) to fluctuating domain walls
forming a ``loop soup''. Specifically, our local Hamiltonian attaches gapless
spin chains to the domain walls of a triangular lattice Ising antiferromagnet.
The macroscopic degeneracy between antiferromagnetic configurations is split by
the Casimir energy of each decorating CFT, which is usually negative and thus
favors a short loop phase with a finite gap. However, we found a set of 1D CFT
Hamiltonians for which the Casimir energy is effectively positive, making it
favorable for domain walls to coalesce into a single ``snake'' which is
macroscopically long and thus hosts a CFT with a vanishing gap. The snake
configurations are geometrical objects also known as fully-packed self-avoiding
walks or Hamiltonian walks which are described by an $\mathrm{O}(n=0)$ loop
ensemble with a non-unitary 2+0D CFT description. Combining this description
with the 1+1D decoration CFT, we obtain a 2+1D theory with unusual critical
exponents and entanglement properties. Regarding the latter, we show that the
$\log$ contributions from the decoration CFTs conspire with the spatial
distribution of loops crossing the entanglement cut to generate a ``non-local
area law''. Our predictions are verified by Monte Carlo simulations.
