Why do you need a relationship field? Can’t you directly use the regions taxonomy with your CPT for the linking to happen?
Now, the filtering of posts by regions can be done easily using a filtering plugin like FacetWP (paid), Search & Filter (free). In order to group the results, plugins like FacetWP have a sort-by feature which can be tweaked to group by your taxonomy. Otherwise without a plugin, it should be a complicated and lengthy code.
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