Product catalog in views that is hierarchical but not mutually exclusive
I would try to achieve this with an exposed filter in views, 'Has taxonomy terms' or 'Has taxonomy terms (with depth)' for example. Use this as an exposed filter (dropdown with multiple select) in a page display. 
In the advanced settings in views you can choose to expose the form in a block. You can add this to the left sidebar and activate 'autosubmit'. So everytime they change something the results get filtered.

Since dropdown is not that nice, I would try on basis of this post https://stackoverflow.com/questions/9208997/drupal-7-creating-a-taxonomy-filter-using-checkboxes to change it to checkboxes.