It seems to be related to ticket #8673, which is marked as wontfix. A workaround is posted in that ticket, but the workaround only works for GET types, not POST. This functionality is important to me.
I would either like jQuery UI Tabs to support sending POST data again using ui.ajaxSettings.data, or a better workaround developed and published. Either way, a note should be added in the API documentation so others won't need to waste hours on this like I did.