Update: I created a new sidebar, and feel like I am really close to a solution, but missing a small piece of the puzzle. The new sidebar is structured like this:
register_sidebar(array(
'name' => 'Tabbed Widget Area',
'id' => 'tabs-widgets',
'description' => 'jQuery UI tab enabled widget area',
'before_widget' => '',
'after_widget' => '',
'before_title' => '<ul>',
'after_title' => '</ul>',
));I am having trouble figuring out how to make the markup for the sidebar match the necessary markup for tabs. I know that all of the widget titles need to be wrapped in a ul, and then each one separately in a li. But I can't figure out how to do it.