Keep global-mode-string on its own "modeline"?

[Thuna]

There is no good reason for `global-mode-string' to be displayed in
every single window's modeline separately, it takes a lot of space, and
it usually ends up not completely showing when the window is split more
than once.  Would it not be better to keep it in a different location?
It makes sense (to me) to have it at the top of each Emacs frame,
although I'm open to suggestions.

Currently, I get something to that effect with tab-bar-mode by setting
`tab-bar-format' to `(tab-bar-format-global)', which does roughly what I
want it to do, but it looks visibly different from the rest of Emacs to
the point where it's rather off-putting.

I don't know what it would take to implement it (if it's even a good
idea), but I just wanted to throw it out there.