I want to add tabs to a custom form in the backend.

There are a couple of great tutorials how to add tabs, like from Inchoo or Erfan Imani.

But I wanted this:

And got this:

The problem is simple - if you know it (costs me an hour and a few hairs):

The tabs are rendered with JS. So they live originally in left, but they are shown in content, to be more specific in the node, defined by:

class My_Module_Block_Adminhtml_Project_Edit_Tabs extends Mage_Adminhtml_Block_Widget_Tabs {

    public function __construct()
    { 
        // ...
        $this->setDestElementId('project_form');
    }
}

And this id needs to be the same as in

class My_Module_Block_Adminhtml_Project_Edit_Form extends Mage_Adminhtml_Block_Widget_Form
{
    protected function _prepareForm()
    {
        $form = new Varien_Data_Form(
            [
                'id'     => 'project_form',
            ]);
    }
}

Hope this helps anyone.