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The following command should not produce any output but it does:
 $ ./python >/dev/null -c 'import subprocess as S, sys; S.call([sys.executable, "-c", "import sys; print(42, file=sys.stderr)"], stderr=S.STDOUT)'
Its stdout is redirected to /dev/null. It starts a subprocess with its
stderr redirected to stdout. See "Redirect subprocess stderr to
stdout" [1] on Stackoverflow.
[1] http://stackoverflow.com/questions/11495783/redirect-subprocess-stderr-to-stdout
I've uploaded a patch that fixes the issue on POSIX.
Please, run the provided test (in the patch), to see whether the code
should be fixed on Windows too (it might work as is there).
No documentation changes are required.
Please, review.

Okay, dumb question: Is there a reason the Windows code explicitly initializes c2pwrite in the "stdout not passed" case, while the Linux code leaves it as -1? Windows doesn't look like it would have the problem (because c2pwrite is always set to a non-default value), and it seems like the fix for Linux could just mimic the Windows approach; the code that sets errwrite wouldn't change, but instead of a "pass", when stdout is None, we'd explicitly set it to os.STDOUT_FILENO, and the stderr=subprocess.STDOUT (stdout unset) case would work automatically, and the code would be more similar.
Haven't explored the negative consequences of that change, if any.

Josh, on Windows, if at least one standard stream is replaced; all three
hStdInput, hStdOutput, hStdError handles are provided
(all-or-nothing). On POSIX, standard streams stdin (0), stdout (1),
stderr (2) are always inherited from the parent. Each stream can be
manipulated independently. c2pwrite=-1 is different from providing
c2pwrite=1 (STDOUT_FILENO) explicitly e.g., set_inheritable() call is
made after the fork() in the latter case.
My patch leads to dup2(fileno(stdout), STDERR_FILENO) when stdout is
None and stderr=STDOUT on POSIX i.e., it redirects stderr to the
inherited stdout (like 2>&1 in the shell). It has no effect otherwise.
sys.__stdout__ is used so that the call fails sooner without fork() if
fileno(stdout) is not a valid file descriptor when python initializes
(daemon, GUI). __stdout__-based solution doesn't support a case when
fileno(stdout) is changed later in the program e.g., using freopen() on
some systems.

Updated the patch to address vadmium's review comments.

I think this patch is pretty good. I will try to commit it in the next few days.

New changeset 642933771fa5 by Martin Panter in branch '3.5':
Issue #22274: Redirect stderr=STDOUT when stdout not redirected, by Akira Li
https://hg.python.org/cpython/rev/642933771fa5
New changeset 5979e7aadd59 by Martin Panter in branch 'default':
Issue #22274: Merge stderr=STDOUT fix from 3.5
https://hg.python.org/cpython/rev/5979e7aadd59
New changeset 5f46ecaf8c6e by Martin Panter in branch '2.7':
Issue #22274: Redirect stderr=STDOUT when stdout not redirected, by Akira Li
https://hg.python.org/cpython/rev/5f46ecaf8c6e 

Thanks for the patch Akira.