Authentication doesn't work with caching enabled in Drupal 5

Because of the following checks introduced in Drupal 5:

function sess_read($key) { 
  ...

  // Handle the case of first time visitors and clients that don't store cookies (eg. web crawlers). 
  if (!isset($_COOKIE[session_name()])) { 
    $user = drupal_anonymous_user(); 
    return ''; 
  } 

  ..
} 

function sess_write($key, $value) { 
  ...

  // If the client doesn't have a session, and one isn't being created ($value), do nothing. 
  if (empty($_COOKIE[session_name()]) && empty($value)) { 
    return TRUE; 
  } 

  ...
}

cookie-less authentication doesn't work when caching is enabled.