Hi Pete,
I’ve been working with AHB for 20 years as a hobbyist/sideline beekeeper in Brazil (250 hives). We never treated a single one for Varroa. What we observe is a variable level of tolerance. While the majority of hives thrives, some dwindle and collapse. We try to eliminate those “less tolerant” queens as soon as possible.
For years we have been told that AHB are excellent because of their rusticity. However, high swarming ratios, nasty behavior and average productivity are a reality. But this works for the great mass of beekeepers. As they lack information on bee husbandry, the sum of rustic bees and exuberant nectar sources guarantees the honey crop. If one would give them “Italian queens” for instance, they wouldn’t last the first year. 
Fairly recently the introduction of new genetic material has become more popular. Buckfast, Italian (Starline, Naveiro), Carniolan and Caucasian lineages area available with some queen breeders (most illegally introduced). Anyway, these virgin queens when open mated with AHB drones result in highly productive bees. They are still far from F0s in terms of defensive behavior, but way better than “native” AHBs. I quote native because this are the bees currently found in nature. Regarding Varroa tolerance, the results are again variable with these hybrids. 
Funny thing is we cannot improve AHB bees through genetic crossings since we don’t have pure genetic material to breed from (there is no “model or reference” AHB lineage). We can only select them for better traits. For example, AHB selected for propolis harvesting are more productive than non-selected hives (Brazil produces tons of green propolis). But that is common knowledge (selecting better traits). This is the reason why people are bringing “European” bees in again to try to reach a middle term between “European” and AHB bees. Maintain those F0 breeders is not that easy here… constant checking and protection from invading AHB queens are necessary. Unsurprisingly, only instrumental insemination works to keep these lineages pure.

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