This works:

use IO::Handle;
use strict;
use warnings;

my $x= IO::Handle->new();
$x->fdopen(fileno(DATA), "r");

my $y= IO::Handle->new();
$y->fdopen(fileno(DATA), "r");

my $pos= tell *DATA;

my $pid= fork();
die unless defined $pid;
if ($pid == 0) {
  seek $x, $pos, 0;
  print "Start c\n";
  while (<$x>) {
    print "c: $_"; sleep 1;
  }
  print "stop c\n";
}
else {
  seek $y, $pos, 0;
  print "Start p\n";
  while (<$y>) {
    print "p: $_"; sleep 1;
  }
  print "stop p\n";
}
__DATA__
a
b
c
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e
[download]

Simply reposition the filepointer…