|logsig Venture-Captain, Washington—Seattle
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There are no disallowed organizations.
However, keep in mind that the usual purpose of Secondary Initiation is not simply to be a member of some organization but to enable taking some character option (e.g. a feat) that has membership in that organization as its access condition or prerequisite. Practically speaking, it isn't the organization itself that might be disallowed, but the character option that it would enable.
So, you should first identify what character option you are interested in taking that has membership in an organization as an access condition or prerequisite. Then, check its (a) rarity and (b) availability, to see whether it would be possible for you to take it. That will help inform your decision on whether to spend Secondary Initiation to become a member of that organization.
- If the character option has the Rare trait, then regardless of anything else, you need a specific boon*.
- If it has the Uncommon trait and there are no access conditions (i.e. organization membership is a prerequisite and not an access condition) and nothing in the book, the PFS Guide or the Character Options page grants you access, then regardless of anything else, you need a specific boon*.
To check availability, go to the PFS2 Character Options page, open up the section for the book that contains the option you want, and see if there are any availability restrictions on that option. (Or if you use Archives of Nethys, PFS availability is shown using symbols.)
- If the option you want has Restricted or Limited availability, then regardless of anything else, you need a specific boon*.
The Uncommon feat Eye of the Arclords (from World Guide) has the access condition "You are a member of the Arclords of Nex". However, we see that this feat is of Limited availability. That being the case, there would be no mechanical benefit in spending Secondary Initiation to become a member of the Arclords of Nex, as you would not be able to take that feat without a specific boon for it*.
* Note that the need for a specific boon does not necessarily mean that any such boon currently exists, will ever exist or that it will be easily obtained, if/when it does exist.