Order tags, but not alphabetically

One way to do it is to store the order of the tags as post meta data, using the action set_object_terms which happens to pass the tags in the order they appear on edit, eg in your “functions.php”:

// Called in admin on updating terms - update our order meta.
add_action( 'set_object_terms', function ( $object_id, $terms, $tt_ids, $taxonomy, $append, $old_tt_ids ) {
    if ( $taxonomy != 'post_tag' ) {
        return;
    }
    // Save in comma-separated string format - may be useful for MySQL sorting via FIND_IN_SET().
    update_post_meta( $object_id, '_tt_ids_order', implode( ',', $tt_ids ) );
}, 10, 6 );

// Reorder terms using our order meta.
function wpse174453_get_the_terms( $terms, $post_id, $taxonomy ) {
    if ( $taxonomy != 'post_tag' || ! $terms ) {
        return $terms;
    }
    if ( $ids = get_post_meta( $post_id, '_tt_ids_order', true ) ) {
        $ret = $term_idxs = array();
        // Map term_ids to term_taxonomy_ids.
        foreach ( $terms as $term_id => $term ) {
            $term_idxs[$term->term_taxonomy_id] = $term_id;
        }
        // Order by term_taxonomy_ids order meta data.
        foreach ( explode( ',', $ids ) as $id ) {
            if ( isset( $term_idxs[$id] ) ) {
                $ret[] = $terms[$term_idxs[$id]];
                unset($term_idxs[$id]);
            }
        }
        // In case our meta data is lacking.
        foreach ( $term_idxs as $term_id ) {
            $ret[] = $terms[$term_id];
        }
        return $ret;
    }
    return $terms;
}

// Called in front-end via the_tags() or related variations of.
add_filter( 'get_the_terms', 'wpse174453_get_the_terms', 10, 3 );

// Called on admin edit.
add_filter( 'terms_to_edit', function ( $terms_to_edit, $taxonomy ) {
    global $post;
    if ( ! isset( $post->ID ) || $taxonomy != 'post_tag' || ! $terms_to_edit ) {
        return $terms_to_edit;
    }
    // Ignore passed in term names and use cache just added by terms_to_edit().
    if ( $terms = get_object_term_cache( $post->ID, $taxonomy ) ) {
        $terms = wpse174453_get_the_terms( $terms, $post->ID, $taxonomy );
        $term_names = array();
        foreach ( $terms as $term ) {
            $term_names[] = $term->name;
        }
        $terms_to_edit = esc_attr( join( ',', $term_names ) );
    }
    return $terms_to_edit;
}, 10, 2 );