hook_breadcrumb() doesn't show breadcrumb
I finally figured out how to alter the breadcrumb. The function needed to return a string 

  function MYFUNCTION_breadcrumb($variables) {
    // On récupère les items de la page s'ils existent
    if (drupal_is_front_page ()) {
      return '';
    }
    if ($item = menu_get_item ()) {
      $title = (isset ( $item ['title'] ) ? $item ['title'] : false);
      $last = '&lt;/div&gt;';
      $separator = '&lt;span class="crumbs-separator"&gt;' . $variables ['crumbs_separator'] . ' &lt;/span&gt;';
      $first_crumb = '&lt;div class="breadcrumb"&gt;&lt;a href="/" title="Display the homepage"&gt;Accueil&lt;/a&gt;' . $separator;
      $key = count ( $variables ['breadcrumb'] ) - 2;
      $final_temp = ($variables ['breadcrumb'] [$key]);
      $final_crumb = ((isset ( $title ) &amp;&amp; $title != "") ? ('&lt;span class="crumbs-current-page"&gt;' . $title . '&lt;/span&gt;') : ($final_temp));
      if (isset ( $item ['page_arguments'] )) {
        foreach ( $item ['page_arguments'] as $argument ) {
          // Keeping the name of the taxonomy in $field_name
          if (isset ( $argument-&gt;field_article_type )) {
            $field_name = "field_article_type";
          } elseif (isset ( $argument-&gt;field_type_remontee_document )) {
            $field_name = "field_type_remontee_document";
          } elseif (isset ( $argument-&gt;field_lien_type )) {
            $field_name = "field_lien_type";
          } elseif (isset ( $argument-&gt;field_type_de_contenu )) {
            $field_name = "field_type_de_contenu";
          } else {
            $middle_crumb = "";
          }
          // retrieving the field and then the link to the taxoomy in the form term/taxonomy/%
          if (isset ( $field_name ) &amp;&amp; $field = field_get_items ( 'node', $argument, $field_name )) {
            foreach ( $field as $taxonomy ) {
              $taxo = taxonomy_term_load ( $taxonomy ['tid'] );
              $nom_taxo = $taxo-&gt;name;
              foreach ( $taxo-&gt;field_taxo_list as $taxo_link ) {
                foreach ( $taxo_link as $link ) {
                  $lien_taxo = $link ['value'];
                  $lien_taxo = (substr ( $lien_taxo, 0, 1 ) == "/") ? substr ( $lien_taxo, 1 ) : $lien_taxo;
                  $middle_crumb = '&lt;a href="/' . $lien_taxo . '" title="Display ' . $nom_taxo . '"&gt;' . $nom_taxo . '&lt;/a&gt;' . $separator;
                }
              }
            }
          }

          $crumbs = isset ( $final_crumb ) ? $first_crumb . $middle_crumb . $final_crumb . $last : '';
          // And here is returned a string
          return $crumbs;
        }
      }
    }
  }