  In this paper we establish new integral representations for the remainder
term of the known asymptotic expansion of the logarithm of the Barnes
$G$-function. Using these representations, we obtain explicit and numerically
computable error bounds for the asymptotic series, which are much simpler than
the ones obtained earlier by other authors. We find that along the imaginary
axis, suddenly infinitely many exponentially small terms appear in the
asymptotic expansion of the Barnes $G$-function. Employing one of our
representations for the remainder term, we derive an exponentially improved
asymptotic expansion for the logarithm of the Barnes $G$-function, which shows
that the appearance of these exponentially small terms is in fact smooth,
thereby proving the Berry transition property of the asymptotic series of the
$G$-function.
