Discovered by Daniel Glen:
If you use the multi-desktop utility Spaces, there is a peculiar interaction with X11 programs. In particular, the 'hints' or 'tooltips' from AFNI in Space #1 (say) will popup when the cursor moves over the corresponding screen location in Space #2. This is a little confusing, since you get an AFNI hint when AFNI itself isn't visible!
As far as I know, there is no way around this problem at present, short of simply minimizing to the Dock all AFNI windows before going to a different Space. That, of course, is somewhat antithetical to the reasons people like multi-desktop setups.