So it's possible that my first few attempts didn't have a matched build of Python on the host - the cross-build certainly relies on mixing the installed runtime with the source stdlib, so that could have been an early cause of issues. I can only suspect that it was a factor in all the earlier reports too. Maybe we should detect this situation and fail faster?

The final problem in my case is that the multiarch paths conflict with the cross-compiling paths. The change below to configure_compiler() got me through a full build:

         if CROSS_COMPILING:
             self.add_cross_compiling_paths()
-        self.add_multiarch_paths()
+        else:
+            self.add_multiarch_paths()
         self.add_ldflags_cppflags()

I have no idea whether this is a safe change in general. Anyone able to weigh in?