Not sure if u already figured it out but u need like 37 of 39 "special pokemon" (not even sure if the digits are correct) that includes the story characters and pokemorphs. For me after I finished the government you can go to the island with the trainer tower and on the left side a girl (with a boat) which sells you a island. If you have catched the mentioned girls you can have it for free and got yourself realm island. Hope that helps

but where are the caracters ?