Due to a bug in TestFlight users who signed up for the Rainway for iOS Beta are unable to participate if they chose not to share their real email. Hidden emails which use privaterelay.appleid.com are not detected by Apple as valid.
To reinitiate the beta sign up process you first need to remove Rainway’s association with your Apple ID.
First go to appleid.apple.com/ and login to your Apple ID.
Next scroll down and find the Security section and manage your “Apps & Websites Using Apple ID”
Finally select “Stop Using Apple ID” which will ask you to confirm. Do so, and your Apple ID will no longer be tied to your Rainway account.
After that you may try signing up for the Rainway for iOS beta again.