"Would the drone mothers have to be
 sisters to queen mothers to get that level of inbreeding?"

All queens are diploid.  That is all queens have two copies of each chromosome.  There is a location on one of the pairs of chromosomes that determines sex.  If both of the pair of chromosome have the same thing at this sex determination location you get a diploid drone.  But, if what each individual chromosome has at this location differs between the pair then you must have a female.  Thus, all queens must differ at this location or they would not be queens.

No matter what the level of inbreeding no fertilized queen can possibly lay 100% diploid drones.  Assume for the moment only a single drone fertilizes the queen and that the sperm (which are all the same from one drone) has the same stuff at the sex determination location as one of the queens chromosomes.  The queen can produce an egg with that chromosome which matches the sperm at the sex location and you get a diploid drone.  Or the queen produces an egg from the other chromosome and this does not have the same stuff at the sex determination location as the chromosome in the sperm and you get a worker.  Extend this thought experiment to a queen that mates with 15 drones in a highly inbred situation and each drone again has a match to one of the queens chromosomes at the sex location and again the queen will lay half workers and half diploid drones from each of those drones she mated with.  When this happens it would not be so easy to spot.  The reason
 is the workers kill those diploid drones soon after hatching and the queen is probably going to lay again in the empty cells giving a fairly normal looking frame of brood most of the time, particularly after the brood is sealed.  Maybe more shotgunned than normal but not half of the cells empty.

If you saw 100% drones from a queen the only way that can happen genetically is if the queen was never fertilized at all.  In which case she will lay 100% drone eggs.  All those drones will be normal haploid drones even thou the egg was laid in a worker cell.  That queen is trying to lay worker eggs but has no sperm to fertilize the eggs.  Unfertilized queens do happen.  I had two of them from one batch of queen cells a couple of years ago.  Both of them were very slow to lay at all.  Other queens from the same batch of cells were fine.   I suppose it might also be possible for a queen to mate normally but have some abnormality such that she stored no sperm and could not lay fertilized eggs.  Or perhaps mate with all drones that were not producing viable sperm.  I doubt if any of these possibilities has anything at all to do with inbreeding.  My limited experience with 100% drone laying queens is they do not last long.  In both the cases I experienced
 there was no queen in the nuc within two or three weeks.  They were not even replaced by laying workers.  My sample was small and my results may not always be the case.

Dick


" Any discovery made by the human mind can be explained in its essentials to the curious learner."  Professor Benjamin Schumacher talking about teaching quantum mechanics to non scientists.   "For every complex problem there is a solution which is simple, neat and wrong."  H. L. Mencken

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