use threads;
use Thread::Queue;
...
my $Q = new Thread::Queue;

sub logmsg {
  // added thread id to messages.
  $Q->enqueue( threads->tid . ": @_"; );
}

async {
    open my $log, '>', 'logfile' or die $!;
    while( $Q->dequeue ) {
        my $now = time;
        my $msec = int( ( $now - int($now) ) * 1000 );
        my ( $sec, $min, $hour ) = localtime(time);

        sprintf $log "%02d:%02d:%02d:%03d %s\r\n", $hour, $min, $sec, $msec, $_; 
    }
    close $log;
}

... 

logmsg( ... );

...

$Q->enqueue( undef ); // to terminate logging thread.