It's not possible without coding a custom solution.
While thinking about a technical solution, it occurred to me that there is a logical problem in any case.
Posts can have multiple taxonomy terms assigned.
It may be that in your particular case you are only assigning a single term to the posts (e.g. only one region), but by design it is possible and common to assign multiple terms.
In which case if you were to visit a post which had both the region terms "south" and "west" assigned, what should the URL be?
site.com/south/post-name/ or site.com/west/post-name/
Now, WordPress lets you use the dynamic %category% placeholder when constructing custom URLs for posts, but there is no equivalent %my-custom-taxonomy% that you can use when setting custom URLs for your post type in its options.
To do so requires coding your own rewrite rules where you make your own placeholder such as %my-custom-taxonomy% and use it in your rewrite rules.
