While I'm not completely sure of what you want, it appears to me you are looking for something like a "datawaiting()" call instead of an end-of-file.
Here's a (somewhat) minimal change to your program with the end-of-file replaced :)

#!/usr/bin/perl

use diagnostics;
use strict 'subs';
use strict 'refs';
use Socket;
use IO::Handle;

my $child;      #filehandle to child process
my $parent;     #filehandle to parent process
my $pid;        #Process ID of child process

socketpair($child, $parent, AF_UNIX, SOCK_STREAM, PF_UNSPEC)
                                                or die "socketpair: $!
+";
$child->autoflush(1);
$parent->autoflush(1);
if ($pid = fork()) {                    #parent
    close $parent  or die "close: $!\n";
    sleep 1;
    print STDOUT time%1000, ": polling child\n";
    if (not datawaiting($child)) {                        ##### CHANGE
+D
        print STDOUT time%1000, ": no input from child yet\n";
    } else {
        my $line = <$child>;
        chomp $line;
        print STDOUT time%1000, ": received <$line>\n";
    }
} else {                                #child
    die "cannot fork: $!"  unless defined $pid;
    close $child or die "close: $!\n";
    print STDOUT time%1000, ": child started\n";
    sleep 5;
    print {$parent} time%1000, ": child printed\n";
    close $parent  or die "close: $!\n";
    exit;
}

sub datawaiting  ####### CHANGED see perldoc -f select
  {
  my $rin = my $win = my $ein = '';
  vec($rin, fileno(shift), 1) = 1;
  return scalar select my $rout = $rin, my $wout = $win, my $eout = $e
+in, 0;
  }
[download]

Outputs:

277: child started
278: polling child
278: no input from child yet
[download]

Is that what you were looking for ?

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