I think what you’re trying to achieve is not easy. When a tag is selected, the user is directed to a new page.
So keeping the user on the same page and changing everything else just to keep the tag selector open seems like a lot of work.
What is the real problem you want to solve? There may be easier alternatives.
Most of the popular methods are covered in this section of the theme developer’s guide. You can also read about those and all the other ones in the repository here