Living systematic reviews (LSRs) are an increasingly common approach to keeping reviews up to date, in which new relevant studies are incorporated as they become available, so as to inform healthcare policy and practice in a timely manner. While journal publishers have been exploring the publication of LSRs using different updating and publishing approaches, readers cannot currently assess if the evidence underpinning a published LSR is up to date, as neither the search details, the selection process, nor the list of identified studies is made available between the publication of updates. We describe a new method to transparently report the living evidence surveillance process that occurs between published LSR versions. We use the example of the living Cochrane Review on nirmatrelvir combined with ritonavir (Paxlovid) for preventing and treating COVID-19 to illustrate how this can work in practice. We created a publicly accessible spreadsheet on the Open Science Framework platform, linking to the living Cochrane Review, that details the search and study selection process, enabling readers to track the progress of eligible ongoing or completed studies. Further automation of the evidence surveillance process should be explored. © Author(s) (or their employer(s)) 2023. Re-use permitted under CC BY-NC. No commercial re-use. See rights and permissions. Published by BMJ.