ANBU members were picked based on skill and intellect. Race, rank, sex, point of origin, etc., was not factored into the selection process. All ANBU are handpicked by the Kage of the village. Most of the time, shinobi were picked for their rare jutsu or kekkei genkai. That does not make it impossible for someone else to join.
That being said, this isn't Naruto. Therefore, a system like the SLO version of ANBU will not be the same as Naruto's ANBU. In order to be smart about this, I think that there should be a position under the Kage that directly controls the ANBU (Like Danzo but maybe not as evil?). He/She will have a test/application for 99% of the candidates for ANBU. Along with the test, they will have the meet certain requirements. Some of these could be (And these are examples) that the shinobi was born in the village to parents who were also born in the village (This ensures loyalty most of the time), the shinobi be a certain age (like 16 minimum), and that the shinobi be at least a Jōnin in the village ranking system.
Now, notice that I said 99%? Well, the remaining 1% that are not selected via the normal process are selected directly by the Kage. These are people that are picked based on skill, and skill alone. Rank, age, point of origin, etc., will not matter in this selection. The selected shinobi will also not undergo the test (if there is one) and is either assigned to the main branch of the ANBU or to the Kage's personal unit.
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