1. Accounts order themselves
A payment says “account A, payment #7, send 5 to B.” Each validator votes for it if it has never seen a different #7 from A. Votes from validators holding more than two thirds of the stake make a certificate, and two conflicting payments can never both get one because any two such groups share an honest validator.
The payment is final once as many validators confirm they recorded the certificate. The receipt holds both sets of signatures.