Filter products by variants' tags
Personally I would use categories for a 'facetted search' filter you describe, as categories are less likely to get out of control. e.g. If tags are seldom re-used then a search for a tag may only produce a single result.

This plugin may help you: https://github.com/iainurquhart/FacetedNav_CraftPlugin

If you're going to use tags then I suggest making use of Craft's search functionality as this will allow you to submit the tag as a search term and also display any entries which contain the word/tag in ANY field of the entries within Craft.