Preheat the oven to 300 degrees F. Line a large baking sheet with parchment and set aside. Spray a large ovensafe bowl or pot with cooking spray and add the popcorn and rice crackers. Place in the oven to keep warm.
In a saucepan fitted with a candy thermometer, combine the sugar, corn syrup, soy sauce, vinegar and 1/3 cup water and bring to a boil over medium-high heat, stirring until the sugar has dissolved. Continue cooking until the mixture reaches 260 degrees F. Stir in the butter.
Drizzle the sugar mixture into the popcorn mixture while stirring with a rubber spatula. Continue stirring until the popcorn mixture is fully coated and cool enough to handle, 2 to 3 minutes.
Scoop 6 large piles of the popcorn mixture onto the baking sheet and mold them into loose balls using oiled hands. Pack them just firmly enough so that they hold their shape but don't pack them too firmly or you might lose a tooth eating them. While they're still sticky, sprinkle with furikake and then let them set at room temp for about 20 minutes.