There are a few things that are probably worth looking into to help with your question. Adobe Target Recommendations is an e-commerce style product recommendation service. It has a catalog (feed) that keeps up with product views/purchases (if integrated) and then you can run activities to highlight certain products based on popularity, site affinity, etc. You can set up HTML templates to render the products. A very typical scenario would be on a product page where you advertise similar products off to the side. Or on a search page where you highlight the products that you last viewed. I am not aware of it integrating directly with a search page and it re-ordering. That sounds more like Search&Promote (if that is still an Adobe product). As for Target proper, it would be more of the delivery engine unless you have AEM and all of your products set up as offers.
To answer the question, I am not aware of any feature that could handle your scenario out of the box. Adobe Target Recommendations would be a place that I would look into a bit further to see if it helps.
