I worked with a group of people once who just didn't care about timezones. All (ok, most) clocks were set to local time, but with some random timezone, about half of the time it was GMT, and 2/3 of the remainder were our actual timezone, but the rest were totally random. I had to write a "figure out what the time actually is" routine in my code because it was such a pervasive problem.
I ended up doing kinda the same. Our API already had a method to retrieve the current unixtime on the server, which I used, together with the user-set timezone, to figure out the UTC offset that the user actually meant to set by adjusting their clock.
