Regular Drupal 7 site users do not see new tabs added by hook_menu_local_tasks_alter
You have to use hook_permission

So in your module file something like this will do the job. (taken from the hook's page).

function hook_permission() {
 return array(
  'administer my module' =&gt; array(
  'title' =&gt; t('Administer my module'),
  'description' =&gt; t('Perform administration tasks for my module.'),
  ),
 );
}


When you do that you will get a new entry in your permissions page (admin/people/permissions). Enable that for the anonymous users and it will show up ok.