Laying workers
Updated
Multiple eggs in a cell, eggs stuck to the cell walls rather than standing in the bottom, and drone brood in worker-sized cells. The colony has been queenless long enough that workers have started laying, and it is one of the harder holes to climb out of.
How to know
The multiple-eggs sign is the clearest, but a new queen can briefly lay two to a cell, so do not convict on that alone. The reliable combination is scattered drone brood in worker cells, producing domed cappings across the middle of the nest where flat worker cappings belong, plus eggs placed on the cell wall because a worker's abdomen is too short to reach the base.
The colony will also be noticeably testy, and there will be no queen and no queen cells worth the name.
Why it happens
Queen substance and open-brood pheromone together suppress worker ovary development. Remove both for long enough, typically a few weeks, and some workers begin laying. Their eggs are unfertilized, so every one becomes a drone, and a colony producing only drones is a colony winding itself down.
It is nearly always a consequence of a queen failure that went unnoticed, which is the argument for checking for eggs on every inspection rather than only when something looks wrong.
Why the usual fix fails
Introducing a new queen to a laying-worker colony usually ends with her killed. The workers behave as though the colony is queenright, because chemically it partly is, and they treat her as an intruder.
The old advice of carrying the hive some distance away and shaking all the bees out rests on the idea that laying workers cannot fly home. They can. It sometimes works and it is not reliable, and it is worth knowing that before you spend an afternoon on it.
What actually works
Repeated frames of open brood. Give the colony a frame of eggs and young larvae from a healthy hive every week for two or three weeks. The open-brood pheromone suppresses the laying workers, and the colony may raise its own queen from the eggs you provided.
The more honest answer for a small operation is often to stop. A laying-worker colony is usually dwindling, mostly old bees, and the effort of rescuing it is better spent combining those bees into a queenright colony over newspaper. You lose the hive as a unit and keep the bees, which is the better trade.
This is the case for running at least two colonies. A spare frame of eggs solves several problems, and you cannot borrow one from yourself.