In what cases are you able to start a postdoc without a "conferred" PhD?

If the postdoc requires "degree in hand" by a certain date, then there's little flexibility. The actual degree must be conferred by the university.

That being said, a good number of postdocs instead require that all of the requirements for the PhD be completed before a certain date. This usually means that that the dissertation has been deposited, if not defended. On the extreme end of flexibility, some will simply take the word of the dissertation advisor that the dissertation will be finished by the start date.

Some postdocs will even allow you to start without a degree in hand if all of the requirements for the degree have been satisfied. Some will be satisfied with a letter from the advisor, others will require a letter from the registrar. How strict they will be may be correlated to how many times they've been burned in the past with non-completers.

I don't think there are stats are on how many postdocs (and in which fields) are strict vs. flexible. What I think you'll find is that once a flexible department has been burned by a "postdoc" who never finished their PhD, then they shift to being more strict.

tl;dr: Ask the postdoc program if they are willing to be flexible and if so, which documentation they require.

Most postdoc offers come with the condition that the candidate will have a PhD by the time that they start the job. In many cases showing the PhD is part of the initial paperwork.

If you want to start a postdoc with a document saying that all requirements have been met and the degree will be formally conferred at such and such a date, I think the only way to proceed is to ask the institution whether this will be acceptable. I don't think you can predict the answer by looking through the nuances of the job ad.

I was once asked by someone who had accepted a postdoc offer at my university (UGA) whether or not she could start the position with such a document. She came from an institution that only formally conferred degrees once a year. From my perspective, the key point is that I had no idea what the answer was, but I brought it up with my colleagues and the department head, it went further up in the university, and the answer came back: OK, so long as all requirements have been met. (In the end the candidate decided to push harder and actually get the degree awarded in the summer before arrival.)

I have also seen a postdoc -- at a very prestigious American university -- started by someone who had not written her thesis by the time of arrival. The next semester she accepted a second, semester-long postdoctoral position and only around the end of the spring semester did I learn that she was still writing up her thesis. (It worked out fine for her, and today she is a well-known strong person in the field.) The idea of starting a postdoc before completing your thesis is one of those things that would simply never have occurred to me, but apparently it happens. As with most things, having a very influential advisor couldn't hurt.

(My understanding is that in the humanities, it is relatively common for someone to start a tenure-track job without a PhD. They are then given a certain amount of time to complete their PhD while simultaneously navigating all the difficulties of their new job. This practice -- which is almost begging for trouble, in my opinion, and most of the stories I have heard that start this way end badly -- is all but unheard of in mathematics, because there is usually a postdoc done in between. You should probably have a PhD by the time you finish your postdoc!)

Moral: you really need to ask.

Let me end with a remark which leans towards the obvious: nothing is for sure until it happens, but if you have some specific reason to doubt that you'll have a PhD by the start time of a postdoc, you should bring that up ASAP and ideally before you accept the position in the first place. In the current job market, there is no lack of qualified candidates with PhD in hand.

I, the question asker, ended up accepting a postdoc in Australia. The University offered me to start the postdoc without the degree, but at a slightly lower salary (which would then be raised once I had the official degree). I countered by saying that I was willing to wait a couple of months to start the postdoc with the degree, to get two years at the advertised salary and have some time off. The job ad said the start date was negotiable. The University then gave in; they let me start right away anyway, at the advertised salary, without the degree, as the PI really wanted me to start ASAP. Later I found out the rules which hold pretty much universally in Australia. At Australian Universities, you are allowed to start a postdoc without the degree in hand. Australian Universities have a minimum wage for someone with an official PhD. Therefore, Australian universities are allowed to pay you less than the advertised postdoc wage if you don't officially have the degree, and some PIs/Universities will try to get away with this if they can, to save money. However, they are not forced to pay you a lower wage, so you can negotiate to get the advertised wage. Just thought I'd post a follow-up answer in my specific context.

You might wonder, why do Australian Universities allow you to start without the degree. One of the big reasons is that Australian dissertations go through a form of peer-review by people external to the University (rather than just the thesis committee and a department chair/impartial rep). You can imagine all the delays this could create. Therefore, to avoid having all PhD candidates being unemployed for months/years while their theses go through peer review (and foreign students immediately being deported!) it is generally seen as fine by PIs to start a postdoc as long as the thesis has been submitted for external peer-review. Most PIs will want the thesis written so they aren't working too much on their thesis while doing their postdoc.