I found out what the problem was...IE 11 (eeeech) determined that all sites were "Local intranet" for some bizarre reason. By default, intranet sites in IE 11 are set to run in compatibility mode ... which I guess is such old tech that it doesn't work with Revive's Asynchronous Tag. My fix was to add the following to the head of our site:
<meta http-equiv="X-UA-Compatible" content="IE=edge">
Problem now solved.