The second (and possibly more powerful) part of our experiment involved pushing back on the business. We told the product owner that any stories that didn't have requirements by the second day of the sprint would be taken off of our task board and not done until the following sprint.
We neglected to specify the time on the second day of the sprint by which we needed all requirements, and our PO delivered some at 11 p.m. But he got them to us for all the stories! This was a great result.
At our next retrospective, we went through each story on the whiteboard, and talked about how we felt about each one now. Interestingly, some stories that had a sad face ended up going well, and some with a happy face turned out to be trickier than we had thought. This gave us a better understanding of what we really need to know about each story before we start working on it.
We couldn't do anything directly about our PO being overworked, or about the business people failing to provide information. But we could try this experiment to make our lack of requirements visible, and push back on the business to say, "We aren't wasting our time and yours if we don't have requirements at the start of the sprint." If this experiment hadn't helped, we'd have tried another one.
Have your retrospectives—but do take Linda Rising's advice, and try small experiments. I am betting you will find unexpected ways to improve how you work.