I did some experimenting and it looks like you have to make your own TabbedPaneUI. I first tried the following as I might suspect you did as well:
// didn't change much even though I thought it would
tabbed.setOpaque(false);
tabbed.putClientProperty("TabbedPane.contentOpaque", Boolean.FALSE);
tabbed.putClientProperty("TabbedPane.tabsOpaque", Boolean.FALSE);
Then I tried to create my own TabbedPaneUI) inheriting BasicTabbedPaneUI
MyTabbedPaneUI ui = new MyTabbedPaneUI();
tabbed.setUI(ui);
It was very easy to make the actual tab area transparent but it was more difficult to make the content area so. It seems that UI is using more components to create the internal parts. I didn’t manage to solve the issue entirely but it is solvable if you have the time to create your own TabbedPaneUI.
Peter