authed_menu.tpl.php

<ul>
<?php foreach($links as $link): ?>
<li>
  <?php print render($link); ?>
</li>
<?php endforeach; ?>
</ul>

gistfile1.php

<?php

function radius_theme_theme() {
  $items = array();
  $items['authed_menu'] = array(
    "render element" => 'links',
    'path' => drupal_get_path('theme', 'radius_theme') . '/templates',
    'template' => 'authed-menu',
  );
  $items['unauthed_menu'] = array(
    "render element" => 'links',
    'path' => drupal_get_path('theme', 'radius_theme') . '/templates',
    'template' => 'unauthed-menu',
  );
  return $items;
}

function radius_theme_preprocess_page(&$vars) {
  $vars['settings_menu'] = menu_tree('menu-settings');
  if (user_is_logged_in()) {
    $vars['settings_menu']['#theme_wrappers'][] = 'authed_menu';
  }
  else {
    $vars['settings_menu']['#theme_wrappers'][] = 'unauthed_menu';
  }
}