Recent work has demonstrated the vulnerability of modern text classifiers to universal adversarial attacks, which are input-agnostic sequences of words added to text processed by classifiers. Despite being successful, the word sequences produced in such attacks are often ungrammatical and can be easily distinguished from natural text. We develop adversarial attacks that appear closer to natural English phrases and yet confuse classification systems when added to benign inputs. We leverage an adversarially regularized autoencoder (ARAE) to generate triggers and propose a gradient-based search that aims to maximize the downstream classifier's prediction loss. Our attacks effectively reduce model accuracy on classification tasks while being less identifiable than prior models as per automatic detection metrics and human-subject studies. Our aim is to demonstrate that adversarial attacks can be made harder to detect than previously thought and to enable the development of appropriate defenses.