A mechanical watch is pretty much out of the question then... As much as it pains me to say this; if you are looking for that kind of accuracy you are pretty much looking in the quartz watch arena. What you want is a perpetual calendar if you do not want to set the date every month, they will run until 2100 and at that time they will need an adjustment. So with all this info not taking price into consideration Seiko offers a few quartz perpetual calendars, and so do a number of other companies such as Citizen, Timex, Omega (they make a few nice ones such as the Constellation) and many others.

If you need any more info feel free to ask.