At my work if you work more than 8 hours in a day, they just won't pay you for the extra. It looks bad on reports, and doesn't matter if the job must be completed by midnight you still can't work 12 hours on paper. What you have to do is split off anything you work in overtime, and put it on the next day's time sheet.
Legally, if your time sheets are inaccurate, you open yourself up for all sorts of legal action including significant fines and imprisonment, so what they've done to cover themselves here is require you to sign it as a true and accurate record of your hours worked. (They refuse to pay you if you don't, which in turn is illegal.)
I think I will follow your example, and find myself another job in a business that follows a basic code of ethics.
This is small print on timesheets anywhere I have worked. The point is that they have something that department of labor can see where employees asserted that the hours reported are correct. That's all DOL cares about if they check. It they are coercing employees to falsify time sheets DOL will have strong feelings. Record your hours independently and have those records ready for side by side comparison. After you leave for a better job, consider reporting their illegal practices so that hopefully others will no longer have to put up with it either. It does not help anybody to have a long history of lying on timesheets. ... Is this timesheet rule in writing anywhere? This would halo help a lot. Sorry I'm really interested in this. I'm a manager and I make sure my team records everything they are possibly entitled to be paid for because I'm a reasonable human being. End of rant ;)
I have also noticed, dissapointingly, that some people never seem to ask themseves moral questions. I think a room full of people in suits serves as a heuristic that tells them that someone else has already asked the requisite moral questions. This is partly how you end up with a city of 100,000 expats working for the government of Saudi Arabia, etc. Imagine living there. You could probably go years without anyone ever having a discussion about morality.
I soon found other employment. It pays less, but I sleep a lot better at night.