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This is a good resource, however, it didn't apply in my situation because it describes the manual calibration process, not the auto-cal (which the F0 probably doesn't even have).

I still haven't come across anything that explains in detail how the auto-cal works and precautions one needs to take when it is running. The reference manual section is something like one paragraph and can be summarized as: "You can turn this on and it will calibrate your clock. You can also turn it off."

If I had to guess, it probably does something similar to the manual process, but just in the MCU logic. It's the lack of detail that got me: I basically ran out of things to try on the UART itself and started looking around at other parts of the chip to see what could at least be indirectly related.