Well the phone auth is working correctly , its the registration form which is not working correctly :
http://prntscr.com/owl85k
The ajax call is redirected 302 to groups page somehow .
It could be due to some third party plugin in your website or due to some kind of server security.
Please contact your webhost and ask them if they have any mod security or server security implemented which can block some ajax calls to server when json data sent with html .
And
please try deactivating all your third party plugins at once and again check this issue .