I bet it's a coaster brake just like an old kid's bike from the 60's and 70's.
My allstates brake by pedalling backwards too, but that pulls a cable for regular drum brakes.
That one is probably inside the axle area and is activated by the chain. There should be some sorta stopper thing that sticks out and bolts up to the frame like those old coaster brakes did. Just a guess though.