Cross domain log-in not working

I'm currently working on a new web site using Commerce API and have an SSP page registered as a touch-point for login/register.

With a returning customer, I pass a customer object to my login.ss handler. The hander logs the customer in using shoppingsession logIn(customer) and returns the response redirect URL. I forward the user to the redirect URL and they're now recognised and logged in on both website and shopping domains.

With new customers, I pass registration data as a customer object to register.ss. The script registers a new account using shoppingsession registerCustomer(customer) and returns the response redirect URL. I forward the user to the redirect URL and they're now recognised and logged in on the web domain - but on the shopping domain, isLoggedIn() and isRecognized() both return false.