The achievements dialog drop-down in the top bar shows the recent achievements across all sites. Each entry has a site-specific icon next to it, so you know where it's from. Badges, however, seem to be displayed relative to the site that you're on.
Here is the view from TeX.SE:
Here is the view from Cross Validated:
Here is the view from GIS:
This seems like a bug; the badge icon should also stay with the site.