Phishing is one of the most important security threats in modern information systems causing different levels of damages to end-users and service providers such as financial and reputational losses. State-of-the-art anti-phishing research is highly fragmented and monolithic and does not address the problem from a pervasive computing perspective. In this survey, we aim to contribute to the existing literature by providing a systematic review of existing experimental phishing research that employs EEG and eye-tracking methods within multi-modal and multi-sensory interaction environments. The main research objective of this review is to examine articles that contain results of at least one EEG-based and/or eye-tracking-based experimental setup within a phishing context. The database search with specific search criteria yielded 651 articles from which, after the identification and the screening process, 42 articles were examined as per the execution of experiments using EEG or eye-tracking technologies in the context of phishing, resulting to a total of 18 distinct papers that were included in the analysis. This survey is approaching the subject across the following pillars: a) the experimental design practices with an emphasis on the applied EEG and eye-tracking acquisition protocols, b) the artificial intelligence and signal preprocessing techniques that were applied in those experiments, and finally, c) the phishing attack types examined. We also provide a roadmap for future research in the field by suggesting ideas on how to combine state-of-the-art gaze-based mechanisms with EEG technologies for advancing phishing research. This leads to a discussion on the best practices for designing EEG and gaze-based frameworks.