Radio Clock
Obtain a radio clock with a USB connection for output of current time synchronized by a time code transmitted by a radio transmitter connected to a time standard such as an atomic clock. Similarly, a receiver of GPS signals could also capture the time signal.
The Meinberg Global company, at least, offers several such devices.
Sundial
Position a sundial outside a window. Attach a webcam to the computer in question. Position the webcam in the window. Write an app to interpret the time of day from current image of the sundial. Caveat: Not applicable in Seattle.
Black Box, No Inputs = No Go
More seriously, if you do not trust the system time, you have no access to a time server, and you cannot ask the user for current time, I cannot imagine any solution. You have a black box with no inputs. The radio clock via USB is adding such an input.
Wikipedia has an article on Clock synchronization.