A new converse bound is presented for the two-user multiple-access channel under the average probability of error constraint. This bound shows that for most channels of interest, the second-order coding rate -- that is, the difference between the best achievable rates and the asymptotic capacity region as a function of blocklength $n$ with fixed probability of error -- is $O(1/\sqrt{n})$ bits per channel use. The principal tool behind this converse proof is a new measure of dependence between two random variables called wringing dependence, as it is inspired by Ahlswede's wringing technique. The $O(1/\sqrt{n})$ gap is shown to hold for any channel satisfying certain regularity conditions, which includes all discrete-memoryless channels and the Gaussian multiple-access channel. Exact upper bounds as a function of the probability of error are proved for the coefficient in the $O(1/\sqrt{n})$ term, although for most channels they do not match existing achievable bounds.