wp_list_pages – children in same list

I’ve been searching and experimenting for a solution to an issue I have with wp_list_pages but have come up empty handed.

As it stands, I’m displaying in my sidebar a list of child pages (for the page the user is on) and then in that list any children of children.

This currently outputs as:

<ul>
  <li>Child page 1</li>
  <li>Child page 2
    <ul class="children">
       <li>Child of child page 2</li>
       <li>Another child of child page 2</li>
    </ul>
   </li>
  <li>Child page 3</li>
  <li>Child page 4</li>
 </ul>

I’m trying to add something to my functions file so that children of children are displayed in the same list (not with a separate ul declaration inside a list item) and then given a class.

The ideal output would be:

<ul>
  <li class="secondary">Child page 1</li>
  <li class="secondary">Child page 2</li>
  <li class="tertiary">Child of child page 2</li>
  <li class="tertiary">Another child of child page 2</li>
  <li class="secondary">Child page 3</li>
  <li class="secondary">Child page 4</li>
 </ul>

Any help would be much appreciated.