In unsecured communications settings, ascertaining the trustworthiness of received information, called authentication, is paramount. We consider keyless authentication over an arbitrarily-varying channel, where channel states are chosen by a malicious adversary with access to noisy versions of transmitted sequences. We have shown previously that a channel condition termed U-overwritability is a sufficient condition for zero authentication capacity over such a channel, and also that with a deterministic encoder, a sufficiently clear-eyed adversary is essentially omniscient. In this paper, we show that even if the authentication capacity with a deterministic encoder and an essentially omniscient adversary is zero, allowing a stochastic encoder can result in a positive authentication capacity. Furthermore, the authentication capacity with a stochastic encoder can be equal to the no-adversary capacity of the underlying channel in this case. We illustrate this for a binary channel model, which provides insight into the more general case.