I posted this in the main forum, but I put "Pigeon" in the title, so only, Delia seemed to notice it ;) So, in case it's of interest, I'll post it again here:
Sometimes the bird incubating the egg cracks it (there can be other causes), and the Albumen leaks out. The Albumen is sticky, so sticks to the bird that cracked it. The egg sticks quite firmly to the feathers, especially as the bird might be on the egg for hours without realising it's cracked. When the bird swaps places with it's partner for nest duties, it flies off with the egg stuck to it's feathers, leaving it's partner confused.
Eventually the egg will come loose, as more of the liquid leaks out and the shell cracks open. It will often fall off a bit at a time, but can drop off all at once. Only problem is the bird is carrying a weight that restricts movement and agility, so it's at risk from predation, though most I've seen have survived.
