I think there are plenty of cases where people prefer not sit on in front of their computer today over going to the fitness studio while preferring going to the fitness studio to sitting on in front of their computer tomorrow.

Changing the near to a far frame changes references. I know that not an exact example of the Dutch Book but it illustrates the principle that framing matters.

I don't think it's hard to get people in a laboratory and offer them different food choices to get a case where a person prefers A for B, C for A and B for C.

I think it's difficult to find general model real life cases because we don't use the idea as a phenomenal primitive and therefore don't perceive those situations as general situations where people act normal but see those situation as exception where people are being weird.

I feel like it happens to me in practice routinely. I see options A, B, C and D and I keep oscillating between them. I am not indifferent; I perceive pairwise differences but can't find a global optimum. This can happen in commonplace situations, e.g., when choosing between brands of pasta sauce or somesuch. And I'll spend several minutes agonizing before finally picking one.

I had the impression this happened to a lot of people.

I can't find it anymore, but years ago I found on LW a recording of an interview with someone who had exhibited circular preferences in an experiment.

The Allais paradox is close to being one such example, though I don't know if it can be called "real-life". There may be marketing schemes that exploit the same biases.

A philosophical case where I feel my naive preferences are circular is torture vs. dust specks. As I said here:

I prefer N years of torture for X people to N years minus 1 second of torture for 1000X people, and any time of torture for X people over the same time of very slightly less painful torture for 1000X people, and yet I prefer a very slight momentary pain for any number of people, however large, to 50 years of torture for one person.

If I ever reverse the latter preference, it will be because I will have been convinced by theoretical/abstract considerations that non transitive preferences are bad (and because I trust the other preferences in the cycle more), but I don't think I will ever introspect it as a direct preference by itself.