Interesting issue. Assuming that your neighbors did not recently plant the monster, I suspect that it has been growing slowly for many years where it is currently planted. I also assume that your neighbors took the described action in order to prevent it from further leaning or potentially damaging your house. Though the tying down of it may well reduce the risk of collapse, I understand your concerns. Recourse to your property CC&Rs or Tucson city code may clarify if there is any legal recourse available. There is also a state prohibition against destroying cacti, so perhaps you can convince your neighbor to preclude a claim against his insurance policy (and the concomitant increase in premium) by having it replanted in the desert.
I don't mean to be flippant about this -- but make sure your homeowner's insurance is paid up and that it will address this claim. Talk to your insurance agent about it. It is likely that your neighbor's insurance will be responsible, but that will likely be a resolution between your insurance company and theirs.
This answer does not constitute legal advice and does not and is not intended to create an attorney-client relationship. The law may vary depending on the state in which you reside. It is intended only to give some direction in which to seek assistance.