After renewing our server certificates I tested the configuration by accessing our main webpage via https. Firefox reported that the page wasn’t encrypted. This happens when certain parts of the page, typically images, are accessed over http. In my case this was due to the <base href="http://eis.comp.lancs.ac.uk" /> in the head section of the page. This caused the header image to be accessed via http even though the link was specified as <img src="fileadmin/template/images/title.png" /> with a relative URL.
In the Typo3 backend edit the template as follows.
config.baseURL = http://myserver.net/ [globalString = _SERVER|HTTPS=on] config.baseURL = https://myserver.net/ [global]
Initially I just removed the http:// from the beginning of the baseURL but it seems to be ignored by the browser if the protocol isn’t specified.
Save the template. Don’t forget to clear the cache. On my version of Typo3 there’s a button showing a lightning bolt up in the top right corner for that.
More details about conditionally setting the baseURL at Typo Free.