In a saucepan, combine the brown sugar, water, corn syrup, butter, and cream of tartar (this bit of acid keeps the sugar from crystallizing.) Bring to a simmer over medium heat, stirring frequently with a wooden spoon to dissolve the sugar. When the sugar syrup reaches a boil, stop stirring and swirl the pan around over the heat so it doesn't burn. As the syrup reaches the caramel stage, the bubbles on top will become smaller, thicker, and closer together. Simmer for 10 to 12 minutes, until the caramel coats the back of the spoon. (If so inclined, stick a candy thermometer in the pot and cook until the syrup reaches the soft-crack stage, about 270 degrees F. When you drop a bit of this syrup into cold water, it will solidify into threads that, when removed, are hard yet bendable.) Remove the caramel from the heat and stir in the marshmallows. Fold the marshmallows into the hot caramel so they all melt into a gooey caramel sauce.
The caramel will coat better if the popcorn is hot, so make it while boiling the sugar or pop it first then keep it warm. Put the popcorn in a big bowl and pour the caramel marshmallow mixture over the popcorn. Gently fold with a wooden spoon to coat each kernel. Grab handfuls of the popcorn and squeeze it together to form popcorn balls; make them about the size of a tennis ball. Put the popcorn balls on a sheet pan lined with waxed paper to cool completely.