Office 2010 requires product key after every reboot

I have been running a Technet subscription registered version of Office 2010 for quite a while. Recently I tried to set up a trial of Office 2013 on this PC, which ended up in disaster. I won't go into details, but suffice it to say I had to eventually abandon the attempt. The problem is that now every time I start an Office 2010 application, it asks for a product key, and then goes through the lengthy activation process (successfully.) I have tried the obvious - uninstalling and reinstalling the Office 2010 product, creating a new user profile, I have even used Microsoft's own supposed "Fix-It" for doing a complete uninstall, and also Revo's uninstaller (found 4000 registry enries left behind after Microsoft's own uninstall process had completed!) None of these helped. Clearly there are tenacious registry entries that nothing will remove, because even after going through these uninstalls, the reinstall picked up all the old e-mail accounts and pointed to the right .PST file on another drive.
It is extremely irritating to have to continually go through the activation process. Any ideas how I can fix this?