I tried your script using File::Tee by salva. It responded pretty much as BrowserUk said tee would. Here's the code, minus the print statements:

#!/usr/bin/perl

use strict;
use warnings;
use File::Tee qw(tee);

tee('STDOUT', '>>', 'stdout.txt');

$| = 1;
my $i = 0;

$SIG{'INT'} = 'handler_of_int';
sub handler_of_int { 
     close('STDOUT'); 
        exit(0); 
}

while (1)  {
    print STDOUT ++$i, "\n";
    sleep (1);
}
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