Autonomous agents deployed in the real world need to be robust against adversarial attacks on sensory inputs. Robustifying agent policies requires anticipating the strongest attacks possible. We demonstrate that existing observation-space attacks on reinforcement learning agents have a common weakness: while effective, their lack of temporal consistency makes them detectable using automated means or human inspection. Detectability is undesirable to adversaries as it may trigger security escalations. We introduce perfect illusory attacks, a novel form of adversarial attack on sequential decision-makers that is both effective and provably statistically undetectable. We then propose the more versatile E-illusory attacks, which result in observation transitions that are consistent with the state-transition function of the environment and can be learned end-to-end. Compared to existing attacks, we empirically find E-illusory attacks to be significantly harder to detect with automated methods, and a small study with human subjects suggests they are similarly harder to detect for humans. We conclude that future work on adversarial robustness of \mbox{(human-)AI} systems should focus on defences against attacks that are hard to detect by design.