Deploying multisite blogs in a subdirectory with a slug

I currently have a multisite subdirectory installation which is obviously deploying blogs as follows:

mydomain.com/theblogname

Because blogs are being created publicly, I would like to insert a slug before the subdirectory blogs to keep things separate and super tidy, much in the way Etsy does with:

etsy.com/shop/theshopname

I would like:

mydomain.com/site/theblogname

(I did search, but I'm surprised nobody else has asked a similar question) How can I achieve this please?