> (Another note: the C-level errno and the TLS copy should also be
> synchronized when the C code invokes a Python callback.)

What I meant is what should occur when a pure Python function is used
as a callback.  At this point there is already some logic e.g. to
re-acquire the GIL if necessary.  Maybe it needs to grow logic to
optionally copy the C-level errno into the TLS variable at the start,
and at the end copy it back into the C-level errno at the end, for the
cases where the C code expects the callback to be able to set errno.