I have spent quite a while researching this issue. My team isn't quite sure of the cause and need a few ideas. I checked the ULS logs where I saw two different errors. 500.0.64 and 403.0.0. I enabled failed request tracing and got 401.2.5. I have tried adding Content-Security-Policy headers and the Vary header with appropriate values. Even before we put the site on https we could not access it in Chrome. Let me know if you need any more information. I am very limited in what I can post, also can't give you the link to the site (sorry). Thanks in advance for any help.
Note: The site is fully functional in IE11 and FireFox.
Note 2: We can't add any add-ons to Chrome. Fixes that are client specific aren't really very helpful. (Plus we tried the DisableLoopbackCheck registry edits and it didn't help)
It turns out I needed to turn on the Negotiate provider in Authentication in IIS. After adding Negotiate and restarting IIS Chrome worked fine. Hope this helps someone having a similar issue.