Prior to the secret ballot's Australian invention in the 1850's (and world-wide adoption by the 1890's), votes were in no way anonymous, and bribery/coercion was rampant.

If you want to give up on anonymity, I can tell you a great voting system. We'll publish everybody's name in the newspaper with how they voted and give you up to a week to correct it. That will do wonders for accuracy. But tell me, do you really want that?