> Regardless of where the vertical line is, it is now almost impossible to
 > get the <=> handle to appear at all when approaching the vertical line
 > from the left -- I get it once in about fifty attempts. When approaching
 > the vertical line from the right, the <=> handle appears normally (and
 > it now appears normally when the vertical line is all the way to the
 > left as well).

If you can apply the patch I posted earlier (and I attach here again)
you could try the following snippet with emacs -Q:

(let ((i 200)
       (window-on-left (selected-window))
       (window-on-right (split-window-right))
       (lhits 0)
       (rhits 0))
   (tooltip-mode -1)
   (scroll-bar-mode -1)
   (set-mouse-pixel-position (selected-frame) i 200)
   (while (car (setq position (cdr (mouse-pixel-position))))
     (let ((coordinates (coordinates-in-window-p position window-on-left t)))
       (when (eq coordinates 'vertical-line)
	(setq lhits (1+ lhits))))
     (let ((coordinates (coordinates-in-window-p position window-on-right t)))
       (when (eq coordinates 'vertical-line)
	(setq rhits (1+ rhits))))
     (sit-for 0.001)
     (set-mouse-pixel-position
      (selected-frame) (setq i (1+ i)) 200))
   (cons lhits rhits))

Evaluating this returns (8 . 0) here, the cdr of which amounts to the
width of one character on my frame.  So here I have an 8 pixel-wide
corridor entirely in the left window where I am "on the vertical line"
(which occupies virtually the 7 right pixels of the right fringe of the
window on the left).  Evaluating

(let ((i 600)
       (window-on-left (selected-window))
       (window-on-right (split-window-right))
       (lhits 0)
       (rhits 0))
   (tooltip-mode -1)
   (scroll-bar-mode -1)
   (set-mouse-pixel-position (selected-frame) i 200)
   (while (car (setq position (cdr (mouse-pixel-position))))
     (let ((coordinates (coordinates-in-window-p position window-on-left t)))
       (when (eq coordinates 'vertical-line)
	(setq lhits (1+ lhits))))
     (let ((coordinates (coordinates-in-window-p position window-on-right t)))
       (when (eq coordinates 'vertical-line)
	(setq rhits (1+ rhits))))
     (sit-for 0.001)
     (set-mouse-pixel-position
      (selected-frame) (setq i (1- i)) 200))
   (cons lhits rhits))

gets me the same results.  What do you get?  You might have to change
the initial values of `i' to "be in the frame" accordingly.

martin