Turns out there’s a very simple solution.
First of all, create a hidden menu. This is a menu that exists in Joomla, but it is not displayed on the website, because the module for that menu is not published (or it doesn’t exist).
Now, create a search page on the hidden menu. In the Menu Manager for the hidden menu, click New, and then pick Search from the list of menu item types. Configure the search results page.
Save this new menu item. Assign modules for the page, if needed, in the Module Manager. You could also assign a specific template to this page if required, if the template you wanted to use was not the default template for the site. Otherwise, the default template would be used to display this page.
And that’s it!!!
Joomla is smart enough to use this menu item as the display page for search results, when the search is run from any page of the website. This should get your search results displaying in the correct template, with the correct modules.
Thanks to Grumpy Bill Tomczak for the tip!