It is fairly hard to perform such checks. We don't have many situations which are analogous to Newcomb's problem. We don't have perfect predictors and most situations humans are in can be considered "iterated". At least, we can consider most people to be using their 'iterated' reasoning by mistake when we put them in once off situations.
The closest analogy that we can get reliable answers out of is the 'ultimatum game' with high stakes... in which people really do refuse weeks worth of wages.
By the way, have you considered what you would do if the boxes were transparent? Just sitting there. Omega long gone and you can see piles of cash in front of you... It's tricky. :)
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I don't think this is at all true that theism is a uniquely awful example.
What about Ayn Rand's Objectivism? I just ctrl-f'd for that on this page and I'm AMAZED to report that no one else seems to have mentioned this obvious example yet.
Things like dualism are right out, but also non-mystical things like homeopathy.
Seems to me there are lots of simply referencable ideas or schools or thought that rationality cleans up.
Secondly, though, I think that rationality often leads reliably to positions that are just a bit less easy to squirt out in a name or phrase like that. Take... free trade, for instance.
The popular common positions are to argue for or against it (mind-killing politics), but I'd think a person using rationality should reliably come to the more nuanced position that free trade is mathematically proven to be the most optimal, but there are problems in the details of switching to it that need to be addressed (ie, what happens when the safety standards between two areas are different? And while yes, everyone will be better off on average afterwards, what about those people who will be negatively affected by the transition? Do we want some policy for helping them through or what?)
So rationality should lead reliably to similar conclusions in most areas, but the conclusions will often be more complex and nuanced than are commonly squawked back and forth in pop-culture.
Secondly, big parts of these rational positions will be the "I don't know so let's find out" spirit, with the agreement being on how to start looking.