I recently had an issue with Tamper Monkey, jQuery, subdomains. You can view the issue here:
You can try it with this tech test. This is without the subdomain:
This is through the subdomain:
You should get alert dialogs when the page loads and when you click the button.
The fix is to use the Beta version of Tamper Monkey. This latest version fixes the bug. You can download it from here.
Also, just FYI, here is my original thread asking for help: