Allowing users to add themselves to other sites on a multisite network

I’m building some client sites as a multisite network to improve ease of maintenance on the back end. That said, I noticed recently that if someone (say, a customer for Client A) tries to register on Client B’s site on my network, they’ll be told they’ve already registered, but they won’t get added to the site in the network.

For privacy policy reasons, a solution such as WPMU Default User Role is too broad, as I don’t necessarily want to give all my clients access to each other’s customers, but I do want their customers to be able to access and subscribe to other sites on the network if they choose. Is there a plugin that will treat a registration of an existing user as if it were an Add Existing User request executed from the back end?