How does this idea sound?
When you arrive at the voting place, a person checks your identification and makes sure you are at the correct voting place and you haven't already voted. For each contest on the ballot, there is a line of turnstiles you can pass through, one for each contestant (or yes/no/abstain for ballot measures). Once you make your first vote, there is another line of turnstiles for the next contest/ballot measure. This could be behind curtains to keep it almost 100% anonymous other than the person or people watching you to make sure you don't turn any turnstiles more than once. At the end of the day, the MECHANICAL turnstiles' counters are read.