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There's a 90% chance that this issue is caused because you have set $live_site in your configuration.php file and it starts with http instead of https. This tells Joomla! to report that the base URL of the site begins with http, no matter if you're accessing your site over SSL. Fact: $live_site is only required on broken hosts. It's a workaround for servers with ill-configured reverse proxies and broken web server installations (like old versions of IIS) which don't report the site's hostname to PHP. If you've set $live_site, unset it. If your site breaks, set it again, move to a decent host a.s.a.p. (because if something as fundamental as reporting the hostname to PHP is broken you don't want to wait long enough to find out what else is broken on that server) and unset it on the new site.

There's a 9.99% chance that this issue is caused by some .htaccess settings, but you'd see the URL change in the browser's address bar.

There's a 0.01% chance that it's a third party plugin (extremely unlikely!) or another factor that I cannot think of and never observed on any other site. There's always a first time for everything, but this happens to me disturbingly infrequently the last two years.

I can tell you that there's exactly 0% chance of this being a bug, because the URL to your site's root is fetched using Joomla!'s own JURI::base() method, the same one used internally by Joomla! itself. The code can be found in plugins/system/admintools/admintools/core.php lines 155 through 162. JURI::base() does read the value of $live_site, that's why the most likely culrpit is $live_site ;)