Ok, I now pass an empty array to setFilter() if no results are returned.

The alises thing - definitely what I would do were I dealing with raw SQL. Is there a way of doing this when instantiating DOM? This is my code. Having a Name field in both Category and Sector was causing Sector.Name to appear in both columns:

    function getCMSFields()
    {
        $fields = parent::getCMSFields();

        $category_manager = new ManyManyDataObjectManager(
            $this,
            'Categories',
            'Category',
            array(
                'CategoryName' => 'Category Name',
                'Sector.Name'   => 'Sector Name'
            ),
            'getCMSFields_forPopup',
            '',// source filter
            '',// source sort
            'LEFT JOIN `sector` ON `sector`.`ID` = `category`.`sectorID`'
        );
        $sectors = DataObject::get('Sector'); 
        $category_manager->setFilter(
            'Sector.Name',
            'Filter by Sector',
            isset($sectors) ? $sectors->toDropDownMap('Name', 'Name') : array()
        );
        $category_manager->setPermissions(array('only_related')); // add, edit, show, delete, only_related
        $category_manager->setPluralTitle('Categories');
		$category_manager->setUseViewAll(true);
        $category_manager->setPerPageMap(array(9999));
        $category_manager->setPageSize(9999);

        $fields->replaceField('Categories', $category_manager);
        
        return $fields;
    }

(Sorry - it seems my code isn't displaying properly in code view, even though I've converted tabs to spaces..)

Would I be able to set an alias here without passing a custom SQL query to DOM?