What do I do for a tenant reference if I've just sold my house?
I will be in the same situation (although I may be renting out our US house).
I'd suggest getting a record of payments and a payoff statement from your mortgage company and presenting that as evidence of reliability along with some personal references they can contact like past employers, your lawyer, something like that.
I've also been a landlord here in the states and have been on the other end of this. Being able to show that you have a serious amount of cash available to you helps a lot. You can offer them a copy of account statements with accounts numbers redacted as proof of ability to pay.