  We study the leading nonperturbative corrections to the strong-coupling
(ungapped) phase of the Gross-Witten-Wadia (GWW) integral over unitary
matrices, to one-loop order. We compute these corrections directly in terms of
eigenvalue tunneling in a holomorphic presentation of the integral over
eigenvalues. The leading nonperturbative contribution to the partition function
comes from a pair of complex eigenvalue instantons. We show that these are in
fact "ghost instantons", which are extrema of the one-eigenvalue effective
potential on the "unphysical sheet" of the spectral curve and have been
discussed in detail recently by Mari\~no, Schiappa, and Schwick. Further, we
discuss the relationship of these instantons to the Fredholm determinant
expansion of the unitary matrix integral, which has recently become an object
of interest in the computations of BPS indices of supersymmetric gauge theories
and black holes. We find that, after taking the 't Hooft limit, the first
correction given by the Fredholm determinant expansion of the GWW integral
agrees precisely with the leading nonperturbative correction, to one-loop
order.
