Im using DOpus v 12.23 (Build 7655).
I just saw that triggering dualdisplay renames the source tab to the name of the destination tab. In case of interest i can offer a screen capture video which i dont want to share here for privacy reasons.
The only way I can think for that to happen is if you have a script installed which is renaming the tabs, or if you're ending up with the two sides swapped without noticing.
(Which side becomes source when you turn on dual display is an option in Preferences, if that's causing the problem.)