Custom Sort Order for Products in Product Tag Page

I would like to create a custom (manual) sort order for products on a product tag page.

I have a, ‘menu order’ option that lets me add priorities to products, but I need this priority to be specific to this set of tagged products and not affect sorting on other pages.

I would like to accomplish this without a plugin, does anyone know how I may do so?

I have not found any threads on this specific topic, but I apologize if I have been using the wrong terminology and missed an answer somewhere.

The page I need help with: [log in to see the link]