simple, don't use buffered IO, use sysread instead:

select STDERR;
$| = 1;
select STDOUT;
$| = 1;

open(CMD, 'perl b.pl 2>&1|') or die $!;
open(LOGFILE, '>', 'b.log') or die $!;

my $input;
while(sysread(CMD, $input, 1024)) {
    print $input;
    print LOGFILE $input;
}
close(CMD);
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just remember to always use sysread on CMD if you use it once