This will be tricky. I'd honestly uninstall both then get just office 2013 reinstalled. In this case that is what I would recommend sadly.
I agree, I have yet to run both or remove one while the other is installing without some kind of issue.
i did have both installed at one point, deleted 2010 afterward using uninstaller, all ok and it has been a while .I used a third party uninstaller
I guess I'll dig deeper as to why this needs to be done also well done hbhb having both on at the same time is a recipe for disaster. Both Office 2010 and Office 2013 use very similar technology, if I were to describe the differences between each it would be like taking Windows 7 and adding a Service pack to it. They are that similar guys its why the office 2013 repository has so little information as there is really nothing to add besides the windows 8 information. Your on a copy of Windows 7 both copies install the same way if you don't uninstall both it will create a freaking mess. Each copy installs licensing data to the registry, each has the licensing in the officeprotectionplatform in the programdata, each has the exact same folder structure (I didn't even have to look this up as I created the office 2010 repository and helped the office 2010 guys). For once take my word on it backup your activation on office 2013, uninstall office 2010 & 2013. I think that covers all the bases
I think it not a big issue... First uninstall office 2010 using "remove programs" & once uninstall completes, use office 2013 setup to perform repair & that will take care of rest of things.