Our BR, Pheobe, did this when she was little. Her droppings were filled with liquid but has solids in a little clump in the middle. She did it all the time, not when we picked her up. But now she's grown up and perfectly fine, a very mellow good layer!
Birds do indeed have renal systems (kidneys), and they do indeed "pee". What they don't have is a bladder. The pee and poo mixes together in the cloaca, and is usually released at the same time. USUALLY -- not always.