Try using an Unbuffered sys.stderr

cibuildwheel/__main__.py

@@ -220,6 +220,7 @@ def main() -> None:
 
     # Python is buffering by default when running on the CI platforms, giving problems interleaving subprocess call output with unflushed calls to 'print'
     sys.stdout = Unbuffered(sys.stdout)  # type: ignore
+    sys.stderr = Unbuffered(sys.stderr)  # type: ignore
 
     print_preamble(platform, build_options)
 

cibuildwheel/docker_container.py

@@ -57,9 +57,6 @@ class DockerContainer:
             ],
             stdin=subprocess.PIPE,
             stdout=subprocess.PIPE,
-            # output the subcommand stderr to our stdout. Avoids ordering
-            # problems between log messages on stderr and stdout
-            stderr=sys.stdout,
         )
 
         assert self.process.stdin and self.process.stdout