I've been testing this iRule with Internet Explorer, Edge, Firefox and Chrome. All these browsers seem to be working fine. Only Chrome seems to do things a bit differently and is showing a login prompt for a split second, but it's working.
How to use this snippet:
Create a Virtual Server that delivers a webserver that uses Kerberos Authentication. Create APM Access Policy that will perform the fallback authentication. Add this iRule to the Virtual Server that holds the APM access policy to perform the fallback authentication.