I assume that this is a simplified example and that you are trying to understand the issue rather looking for a workaround, but in just in case... you could read the DATA before forking...

use strict;
use warnings;

my @data = <DATA>;

my $pid= fork();
die unless defined $pid;

if ($pid == 0) {
    print "Start c\n";
    for (@data) {
        print "c: $_"; sleep 1;
    }
    print "stop c\n";
}
else {
    print "Start p\n";
    for (@data) {
        print "p: $_"; sleep 1;
    }
    print "stop p\n";
}
__DATA__
a
b
c
d
e
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I admit that I do not know either what the fork/DATA problem is. If you put in a sleep statement before the loop you can control whether the parent or the child reads the DATA segment but this is not helpful either.