Since I can't easily tell the two apart from the command prompt, I don't really how to be sure if anything I put in has worked.

perl yourscript.pl >stdout.txt 2>stderr.txt
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Although I feel local'izing the file handles is cleaner, as suggested by JavaFan, you could alternatively store them. Then re-store them. That way, you'd have STDERR available in the block if you needed it.

my ($out, $err) = (*STDOUT, *STDERR);
eval{
    *STDERR = *STDOUT;

    # pod2html code here
    # ...
};
*STDOUT = $out; *STDERR = $err;
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