I have a similar issue. IIS6 win2003, 2 servers in a round robin.
The site with Window Integrated Authentication (WI) enable. I will get prompted randomly for username and password.
I can hit the site
I get prompted
I can hit Escape (no need to enter the username and pass)
I hit Refresh and it works.
Here is where the random comes in.
Sometimes I can hit Refresh again and it still works, some times it prompts again.
If it prompts again I can start over and hit Escape the Refresh.
If I go to the IP of either server it always work, if I go to the IP of the VIP it is Random.
I do have OneConnect enabled.