By utilizing a non-binding, anonymous straw poll, elected officials can express their true leanings without feeling the political backlash from myriad sources. While such a measure would have to be used as a “non-binding procedural aid” (the Constitution requires a recorded vote if one-fifth of the quorum requests it), an anonymous straw poll can produce surprising results and offer political cover during the debate over the real vote. Oscar Wilde once said that if you give a man a mask, he will tell you the truth. This temporary “veil of conscience” would allow members of Congress to express their true sentiments without crossing their party leadership, political donors, lobbyists and even their own electorate. For one brief moment, politicians can vote for the nonpartisan common good as they truly perceive it.
Link thanks to Dglenn, who also isn't sure whether this would work.
ETA: This isn't about anonymous binding votes, it's about a non-binding straw poll so that legislators can express their opinions without fear of reprisal.