Tags on products vs tags on blog posts. An easy way to duplicate them

OK, so you can use created_term hook to make the product tags copy themselves automatically to post tags.

function ( $term_id, $tt_id, $taxonomy ) {
    if ( 'product_tag' == $taxonomy ) {
        $term = get_term( $term_id, $taxonomy );
        wp_insert_term( $term->name, 'post_tag' );
    }
}
add_action( 'created_term', 'copy_product_tags_for_blog', 10, 3 );

And you’ll have to run a simple loop for tags that are already in DB.