The interesting part is whether M. ibericus queens do actively remove their own genetic from eggs fertilized with builder sperm. Why would they do this?
M. ibericus queens produce ibericus×structor hybrids as infertile female worker ants.
M. ibericus queens produce structor males, so future queens can keep producing the hybrid worker ants.
My guess is, maybe there is some benefit having the workers to be hybrids and not pure ibericus ("hybrid vigor" [1]). So it's worth the effort of keeping the structor males along, to be able to produce the hybrid workers. But I think the pure ibericus genes in the line of queens are in control.
Who is doing the removing at fertilization is interesting mostly in a mechanical sense. The mechanisms that worked against it are being suppressed or selected out entirely.
It sure is an interesting case that one ant species is having another species promote their males if one looks at it from a gene perspective. A very weird case of symbiosis.
If I understand it correctly their theory is as follows. Stage 0: 5 million years ago they were the same species. Stage 1: Subspecies. Ibericus and Structor became distinct populations of the same species still capable of mating. Stage 2. Loss of function. Ibericus lost the ability to create their own workers, but as Ibericus and Structor existed in the same places hybrid workers allowed Ibericus to survive despite this. Stage 3: Ibericus learns to clone structor males to live in places where there are no Structors naturally.
Kinda interesting that even though the end result is the same who is considered the parasite is different.
Note that this second scenario doesn't necessarily imply that M. structor males are not some sort of parasites. Actually, because it's detrimental to M. structor males to father only workers (they don't transmit their genes to the next generation), it's easy to imagine selection to remove DNA of the mother so they can be cloned in a fertile caste (males). That being said, maternal DNA elimination prompted by the male genome would be nearly immediately also beneficial to M. ibericus (because having M. structor males at home is good for them), so this maternal DNA removing is something that benefits to both partners, which is why it's not that difficult to imagine that it evolved and has been succesfully selected. When something benefits to both partners of any relationship, there are good chances that it will be selected at some point.