And also show anyone else who was interested that he "voted correctly"? A proper secret ballot requires that nobody--even the voter himself--be able to prove how he voted except in near-unanimous elections (if 120 people voted, and there are 119 votes for John Smith, the person who didn't vote for John Smith knows how everyone else voted). Otherwise, it would be possible for someone to coerce anyone who would be able to prove how he voted, to actually do so.
Well, if you want everyone to know who you voted for, then yes, just give your stub away. It would be alot easier to just tell them though.
The system would be purely optional for the voter to participate in. You don`t want to check, dont take the stub off the ballot.
And as for people buying the stubs , or Union thugs pressuring people to bring them in, that sounds like a crime already on the books, or easily could be. How about making it a felony with a mandatory minimum sentence.... lets say 10-15 years in a federal prison.
But in all honesty, do you ever think this little idea of mine would happen?