  We study the change in the resurgent asymptotic properties of a trans-series
in two parameters, a coupling $g^2$ and a gauge index $N$, as a system passes
through a large $N$ phase transition, using the universal example of the
Gross-Witten-Wadia third-order phase transition in the unitary matrix model.
This transition is well-studied in the immediate vicinity of the transition
point, where it is characterized by a double-scaling limit Painlev\'e II
equation, and also away from the transition point using the pre-string
difference equation. Here we present a complementary analysis of the transition
at all coupling and all finite N, in terms of a differential equation, using
the explicit Tracy-Widom mapping of the Gross-Witten-Wadia partition function
to a solution of a Painlev\'e III equation. This mapping provides a simple
method to generate trans-series expansions in all parameter regimes, and to
study their transmutation as the parameters are varied. For example, at any
finite N the weak coupling expansion is divergent, with a non-perturbative
trans-series completion; on the other hand, the strong coupling expansion is
convergent, and yet there is still a non-perturbative trans-series completion.
We show how the different instanton terms `condense' at the transition point to
match with the double-scaling limit trans-series. We also define a uniform
large N strong-coupling expansion (a non-linear analogue of uniform WKB), which
is much more precise than the conventional large N expansion through the
transition region, and apply it to the evaluation of Wilson loops.
