Interesting. Even duplicating the DATA handle in the child doesn't help.

open my $d2, '<&', *DATA or die $!;
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But opening the script itself again works, using tell to find the position of the DATA section:

#!/usr/bin/perl
use warnings;
use strict;

open my $d2, '<', $0 or die $!;
seek $d2, tell *DATA, 0;

if (fork) {
    while (<DATA>) {
        print "parent: $_";
    }
} else {
    while (<$d2>) {
        print "child: $_";
    }
    exit
}

__DATA__
a
b
c
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