Add the below snippet to functions.php to output the associated tags with a product in the archive pages. Update as neccessary.
add_action( 'woocommerce_after_shop_loop_item', 'woocommerce_product_loop_tags', 5 );
function woocommerce_product_loop_tags() {
global $post, $product;
$tag_count = sizeof( get_the_terms( $post->ID, 'product_tag' ) );
echo $product->get_tags( ', ', '<span class="tagged_as">' . _n( 'Tag:', 'Tags:', $tag_count, 'woocommerce' ) . ' ', '.</span>' );
}