  We show that corner polyhedra and 3-connected Schnyder labelings join the
growing list of planar structures that can be set in exact correspondence with
(weighted) models of quadrant walks via a bijection due to Kenyon, Miller,
Sheffield and Wilson.
  Our approach leads to a first polynomial time algorithm to count these
structures, and to the determination of their exact asymptotic growth
constants: the number $p_n$ of corner polyhedra and $s_n$ of 3-connected
Schnyder labelings of size $n$ respectively satisfy $(p_n)^{1/n}\to 9/2$ and
$(s_n)^{1/n}\to 16/3$ as $n$ goes to infinity.
  While the growth rates are rational, like in the case of previously known
instances of such correspondences, the exponent of the asymptotic polynomial
correction to the exponential growth does not appear to follow from the now
standard Denisov-Wachtel approach, due to a bimodal behavior of the step set of
the underlying tandem walk. However a heuristic argument suggests that these
exponents are $-1-\pi/\arccos(9/16)\approx -4.23$ for $p_n$ and
$-1-\pi/\arccos(22/27)\approx -6.08$ for $s_n$, which would imply that the
associated series are not D-finite.
