My iPhone 5C's time won't move forward unless I'm using my phone

I'm an American student, currently studying abroad in Rome, Italy and over this weekend I went to London. When the time zone on my phone changed, I thought everything was fine, until later that day when I realized that my phone was like 9 hours off the actual time. I tabled the issue, figuring that when I came back to Rome the time change would cause the phone to revert back to normal, however it didn't. I've tried everything; resetting to factory settings, the whole off and on tactic, updating the software, connecting to iTunes and no matter what I do the only way the phone gets set to an accurate time is when I turn the "set automatically" setting in the "Date and Time" section off and on. It'll update to the current time, but then I lock my phone and if I open it two hours later, the phone will operate on the time that I locked it at. I've even tried just keeping the automatic setting "off" but not even that does anything. Please help me haha