They started as small green orbs, and the first thought was limes. But they kept on growing, past the usual size of limes, but remained green. Orange or lemon, perhaps. Short of cutting one down, presumably before being ripe, there was no way to tell.
Well, the cockatoo solved the mystery. They appear to be very unripe oranges - judging by the carcas left behind. And the cockatooes are impatient eaters - there is much edible material left behind, but they seem to tire easily and drop it to the ground. And take another.
They are equally destructive on ordinary tree branches, where they appear to chop off the smaller/newer pieces and drop them on the ground. It's quite common to see them on the footpath when walking under trees.