Since I already did this once I just did a more thorough job; patch is
attached.

PyObject_WriteString() just calls PyFile_WriteObject() which ends up
using PyObject_Print(), so it is was simple to handle those cases.  I
then released the GIL for strings, lists, and ints.  That is enough to
pass Armin's deadlock test.

If the approach I am taking is OK with people and I can go through
Object/*.c and do the proper thing for all the various object types for
fprintf(), fputs(), fputc() and fwrite() (while skipping all of the
debug output stuff) before committing the changes.