I've looked all over and haven't found much regarding this specific situation, and would like some opinions on how to handle this.
Undead and Constructs have specific rules for what to do without a con save. Undead use charisma, constructs treat theirs as 10 for DCs and other statistics that rely on Constitution.
The Construct Caller Eidolon gains the Inevitable Subtype but none of the things that come with that, as per Unchained Eidolon rules. Construct Caller, however, states "A construct eidolon functions as an inevitable eidolon except as noted here. A construct eidolon can be of any alignment. It has no Constitution score and gains bonus hit points appropriate for a construct of its size."
I am having trouble understanding the implications of this. Surely, it can't bleed, right? Well, what about fort saves or disease? With the inevitable gaining immunities to a multitude of effects like disease, poison, drain, etc as it levels (including fort saves at 20), I am inclined to believe that Construct Caller's Eidolon is subject to all fort saves, including ones that thematically make no sense, like bleed.
Of course, adding the construct's clause to treat con as 10 helps in most of these cases, but I just want to know what other's takes on the no con score thing is.