> I'm confused.  What does "there is no redrawing" mean?  That the rest
> of the frame is orderly redrawn or that the rest of the frame is not
> redrawn?

I mean literally. Emacs frame does not redraw properly. I tried make
several window splits after M-x menu-bar-mode. See the attached
screenshot.

I don't even need to change fonts or load emacs theme to trigger this.

> If drawing the Lucid menu bar _is_ responsible for the problem, could
> you try setting its font separately?  See appendix

I tried to run emacs with

emacs -Q -xrm 1.txt

with 1.txt:
Emacs.pane.menubar.font: Courier-12

The result is same.

Best,
Ihor