You need to wrap the while loop in a for loop that iterates once for each client you start:

use strict;
use threads;
use threads::shared; ### I omitted this from my example above initiall
+y.
use Thread::Queue;
use ClarRPC;

my $queue = new Thread::Queue;

rpc('10.15.51.208', '1300', 'ping -n 50 10.15.51.208');
rpc('10.15.51.208', '1301', 'ping -n 10 10.15.51.208');
rpc('10.15.51.208', '1302', 'ping -n 50 10.15.51.208');

for( 1 .. 3 ) { ## Must iterate once for each thread started ###
    while (my $ref = $queue->dequeue) {
        my ($port, @results) = @$ref;
        print "$port : @results\n";
    }
}

sub rpc
{
    my ($ip, $port, $command) = @_;
    async{
        my $connection = ClarRPC->connect($ip, $port);
        my @resp :shared = $connection->rpc(
              'ClarRPCService::system_call', $command
        );
        $connection->disconnect();
        unshift(@resp, $port);
        $queue->enqueue(\@resp, undef);
    }->detach
}
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