I have also experienced this problem.
I am using version 13.10 (32 bit), so decided to disable as many add-ins as possible, as suggested by the moderator.
The results:
1. When only one Word file is open (and one tab showing), editing the file does not result in the * showing. In other words, there is no indication in the tab that the file has been modified.
2. When I open another file (and so another tab) then the * suddenly appears in the first tab. But the * doesn't appear in the second tab. if I open a third tab, then I see the * in the second tab. And so on and so forth.
In addition, the function "Open in new Window) causes more serious problems. It attempts to open the file in a previous version of Word (2010) installed on my PC, which in turn triggers the install routine for Office 2010. I am able to cancel the install, but the file opens in Word 2010, and the new Window is not accessible from Windows. The file is open in Word 2010 though, because it can't be moved, deleted etc. And even though i cancel the Office 2010 install, it has still changed the defaults for doc and docx so i have to reset these as well.
There seems to be something badly wrong with this version of Tabs for Word.
Stephen Yeo