wc-ajax=update_order_review 403 error on multisite

I wanted to give an update to an old unresolved issue that I just ran into myself and was able to solve.

Here is the old topic which only provides a solution for non-multisites: https://www.ads-software.com/support/topic/wc-ajaxupdate_order_review-403-error/

The Problem
When you try to login during checkout you get an error 403 from wc-ajax=update_order_review and cannot finish the checkout.

The Cause
For your multisite you had to fix a common cookie/login problem by adding the following lines (or similar) to your wp-config.php:

define( 'ADMIN_COOKIE_PATH', '/' );
define( 'COOKIE_DOMAIN', 'false' );
define( 'COOKIEPATH', '' );
define( 'SITECOOKIEPATH', '' );

The Solution
You only need one line from the multisite cookie/login fix which doesn’t interfere with WooCommerce:
define( 'COOKIE_DOMAIN', 'false' );