The OP didn't say anything about the two threads communicating with each other through the socket. That wouldn't work. You always communicate with the other end of the socket. If that's what he wants to do, he should create a pair of connected sockets using the builtin socketpair (portable*) or Win32::Socketpair's winsocketpair (Windows only). There's also the option of using a pipe (created using pipe) instead of a socket (although that wouldn't be selectable on Windows).

use strict;
use warnings;
use threads;

use IO::Handle qw( );    # Not needed in Perl 5.12+
use Socket     qw( AF_UNIX SOCK_STREAM PF_UNSPEC );

sub socketpipe {
    socketpair($_[0], $_[1], AF_UNIX, SOCK_STREAM, PF_UNSPEC)
        or return undef;
    shutdown($_[0], 1);  # No more writing for reader
    shutdown($_[1], 0);  # No more reading for writer
    $w->autoflush(1);
    return 1;
}

socketpipe(my $r, my $w)
   or die($!);

async {
   print while <$r>;
};

async {
   print($w "abc\n");
   sleep(3);
   print($w "def\n");
   shutdown($w, 1);
};

$_->join for threads->list;
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* — In Windows, you have to request the creation of AF_UNIX socket pair, but it actually creates an AF_INET socket pair.