  Using the notion of integral distance to analytic functions, we give a
characterization of Schatten class Hankel operators acting on doubling Fock
spaces on the complex plane and use it to show that for $f\in L^{\infty}$, if
$H_{f}$ is Hilbert-Schmidt, then so is $H_{\bar{f}}$. This property is known as
the Berger-Coburn phenomenon. When $0<p\le 1$, we show that the Berger-Coburn
phenomenon fails for a large class of doubling Fock spaces. Along the way, we
illustrate our results for the canonical weights $|z|^m$ when $m>0$.
