When focus is on an html password field, the input languages options are limited to those using the Latin script only. If the language selected is some other language using a non-Latin script, like Greek, before going into the password field, it is automatically switched to one for Latin when focus goes onto the password field. This happens on all browsers I've tested: Safari (8.0.5), Chrome (51.0.2704.103), Firefox (47.0). I've tried it on Mountain Lion, Yosemite and EL Capitan.
I have tried disabling the keyboard shortcut for switching languages but the problem remains.
The option "Automatically switch to an document's input source" mentioned here is not enabled.
The non-english languages are simply disabled when focus is on a password field.
Here is a minimal html document which demonstrates the problem:
<html> <form> First name:<br> <input type="text" name="firstname"><br> Last name:<br> <input type="password" name="password"> </form> </html>