The great collector of Americanisms, Schele De Vere, calls “Anthony Over” an American game and defines it (1872): “A game of ball played by two parties of boys, on opposite sides of a schoolhouse, over which the ball is thrown.”
He localizes it in Pennsylvania, but it was also played in southern Ohio in our childhood, and, if we recall correctly, also in our later youth in the suburbs of New York City.
But why the game carries the name Anthony (or Antony) is a mystery.
Possibly the reference was to St. Anthony who, upon appeal, is credited with aid in finding lost objects, as it was always an object by either of the “two parties of boys” to so throw the ball as to go far over the heads of the opposing party, perhaps to be lost in high grass or weeds.