Continue execution after WP REST API response

One thing you can do in WordPress is to dispatch a non blocking new request to itself, and then resume the response.

Something like this:

$action_name = 'sample_action';
$url = add_query_arg(
    [
        'action' => $action_name,
        'nonce'  => \wp_create_nonce( $action_name ),
    ],
    admin_url( 'admin-ajax.php' )
);

$body = [ 'your data' ];

// trigger a non-blocking request.
wp_remote_post(
    $url,
    [
        'timeout'   => 0.01,
        'blocking'  => false,
        'body'      => $body,
        'cookies'   => $_COOKIE,
        'sslverify' => apply_filters( 'https_local_ssl_verify', false ),
    ]
);

// return rest response and free the request.
return rest_ensure_response( 'success' );

And then, you add the usual callbacks to handle the admin ajax call:

add_action( 'wp_ajax_' . $action_name, 'so_377405_callback' );
add_action( 'wp_ajax_nopriv_' . $action_name, 'so_377405_callback' );

function so_377405_callback() {

    // Don't lock up other requests while processing.
    session_write_close();

    if ( ! isset( $_REQUEST['nonce'] ) || ! wp_verify_nonce( sanitize_text_field( $_REQUEST['nonce'] ), 'sample_action' ) ) {
        wp_die();
    }

    // ... do stuff

    wp_die();

}

Is 3 years too late for a response?