Poll workers will give you a provisional ballot if you absolutely insist (even under false pretenses) but the voter must still follow-through and provide the proper documentation (such as their original voter registration card) to prove that they'd actually registered as a Democrat.
If the chief judges at the polling place are convinced that the voter has the correct documentation, then he/she can verify it, then and there. The judge/s fill in the proper form, and the ballot will be counted during canvassing later on at the dept of elections office.
If no documentation is available at the time, the voter can still fill in the provisional ballot which is saved in a privacy envelope with their name and code number (that's also printed on a tear-off receipt). The judges fill in the form that indicate what the problem was, and what the voter must do in order to correct the problem.
At that point, it's incumbent upon the voter to then return to the dept of elections (usually within ten days) to present whatever proof of eligibility was required, or to present proof that a mistake had been made. Upon the presentation of their truthful documentation AND their "tear-off" receipt (that identifies their provisional ballot envelope) ... if the elections office is satisfied, then their ballot is retrieved, removed from the envelope, and counted.
If the voter is lying or presenting fraudulent documentation, then they open themselves up to prosecution.
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