I'm also facing the very same problem with my iPhone 5C:
-The clock won't change while iPhone is locked - Alarm won't work (obviusly);
-GPS is offset by a few thousand feet;
-Carrier bar only says "searching", but phone calls works normally.
and I tried EVERYTHING possible:
-Restored factory settings;
-Configured as a new iphone;
-Restored from original Back-Up;
-Uptaded all available iTunes and iOS;
-Re-did all the above numerous times;
-Tried DFU mode a few times;
-Prayed;
Then, I went a little deeper, since the warranty was over anyway:
-Opened the iPhone 5C, disconected the battery for a good hour to try to "hard reset" anything - Didn't work;
-Disconected all flat cables from logic board, looked all conections, sensors, and antenas - didn't work;
-Prayed a little more....
Seems that this is a repeating event on iPhone 5C - Never heard about it in any other model.
Am now seriously thinking to change the entirely logic board, but this is way to expensive.....
Has anybody succesfully fixed this issue?!