Reopening STDERR is right, but there is an other solution too, using $SIG{__DIE__} as follows (untested code):

BEGIN{
open LOGFILE, ">>", "/path/to/log" or 
        die "cannot open logfile: $!";
$SIG{__DIE__}= sub {
        print LOGFILE $_[0];
}
$SIG{__WARN__}= sub {
        print LOGFILE $_[0];
}
}
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