Here is the solution in case anyone else is looking for the same thing. It includes the link for the tags, which I’m sure is more useful for most people.
add_action('woocommerce_after_shop_loop_item', 'show_tags', 5);
function show_tags()
{
global $product;
// get the product_tags of the current product
$current_tags = get_the_terms(get_the_ID() , 'product_tag');
// only start if we have some tags
if ($current_tags && !is_wp_error($current_tags))
{
echo '<p class="product_tags">';
//append Tag or Tags
if (count($current_tags) == 1)
{
echo '<span class="product-tags">Tag: </span>';
}
else
{
echo '<span class="product-tags">Tags: </span>';
}
// for each tag we create an item
foreach ($current_tags as $tag)
{
$tag_title = $tag->name; // tag name
$tag_link = get_term_link($tag); // tag archive link
$separator = ' | ';
$tagstrings[] = '<a href="' . $tag_link . '">' . $tag_title . '</a>';
}
echo implode($separator, $tagstrings) . '</p>';
}
}