I don't know whether Anna used this as an illustration, but one way by which I tend to notice myself rationalizing is when I'm debating something with somebody. If they successfully attack my position, I might suddenly realize that I'm starting to defend myself with arguments that even I consider bad or even outright fallacious, and that I've generally gone from trying to discover the truth to trying to defend my original position, no matter what its truth value.
Another example is that I might decide to do or believe something, feel reluctant to explain my reasons to others because they wouldn't hold up to outside scrutiny, and then realize that wait, if my reasons wouldn't hold up to outside scrutiny they shouldn't hold up to inside scrutiny either.
Do you experience either of those?
Anna Salamon and I are confused. Both of us notice ourselves rationalizing on pretty much a daily basis and have to apply techniques like the Litany of Tarski pretty regularly. But in several of our test sessions for teaching rationality, a handful of people report never rationalizing and seem to have little clue what Tarski is for. They don't relate to any examples we give, whether fictitious or actual personal examples from our lives. Some of these people show signs of being rather high-level rationalists overall, although some don't.