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I am looking at the warforged AC calculation from the Integrated Protection feature, looking only at Darkwood Core (unarmored), for a possible warforged barbarian. (I am using the version of the warforged race from the Wayfinder's Guide to Eberron.)

Darkwood Core says:
11 + your Dexterity modifier (add proficiency bonus if proficient with light armor)

What I want: Darkwood Core + Con
What I'm expecting: 11 + Dex + Con

Would this option be able to coincide with the Unarmored Defense for the Barbarian, or is this a case of conflicting Armor Classes?

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Different AC calculations don't stack; you pick which one to use

As Slagmoth pointed out, this general topic has been asked and answered many times by the designers, and an official answer appears in the Sage Advice Compendium:

These [AC calculation] methods—along with any others that give you a formula for calculating your AC—are mutually exclusive; you can benefit from only one at a time. If you have access to more than one, you pick which one to use. For example, if you’re a sorcerer/monk, you can use either Unarmored Defense or Draconic Resilience, not both. Similarly, a druid/barbarian who transforms into a beast form that has natural armor can use either the beast’s natural armor or Unarmored Defense (you aren’t considered to be wearing armor with natural armor).

(The entry goes on to clarify that AC bonuses, like those provided by shields, do stack with AC calculations; it's simply that if your features give you different ways to calculate your AC, you pick one.)

In addition, Keith Baker - original Eberron designer, and primary author of the Wayfinder's Guide to Eberron - has posted an FAQ on his website about the Wayfinder's Guide (which he has continued to update to answer questions). It contains a section about the Integrated Protection feature, which specifically answers this question and references the Sage Advice ruling:

If I’m playing a warforged barbarian, do I get to add my Constitution to my Integrated Protection? What about a warforged monk?

No and no. Integrated Protection and Unarmored Defense are two separate features that set your AC, and the Sage Advice ruling on this is that you pick one—you don’t combine them. If you have your Integrated Protection mode set to darkwood core, you are considered to be unarmored and can thus use Unarmored Defense—but you’re always either using one or the other to determine your AC.

The section also clarifies whether the Integrated Protection feature counts as "armor"/"being armored" for other features. It's a worthwhile read.

In short: you have the choice between two different AC calculations:

or

They don't stack. You simply have the choice between one and the other (presumably you'd choose the one that gives you a higher overall AC).

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Armor Class calculations do not stack

Page 2 of the Sage Advice Compendium clarifies how to calculate armor classes when you have multiple options:

  • Unarmored: 10 + your Dexterity modifier.

  • Armored: Use the AC entry for the armor you’re wearing (see PH, 145). For example, in leather armor, you calculate your AC as 11 + your Dexterity modifier, and in chain mail, your AC is simply 16.

  • Unarmored Defense (Barbarian): 10 + your Dexterity modifier + your Constitution modifier.

  • Unarmored Defense (Monk): 10 + your Dexterity modifier + your Wisdom modifier.

  • Draconic Resilience (Sorcerer): 13 + your Dexterity modifier.

  • Natural Armor: 10 + your Dexterity modifier + your natural armor bonus. This is a calculation method typically used only by monsters and NPCs, although it is also relevant to a druid or another character who assumes a form that has natural armor.

The fun part is that the Warforged entry does not indicate whether their feature counts as Armored or as Natural Armor. I would suspect (and as a DM say it was intended to be the latter allowing barbarian and monk features) without developer guidance it would be up to your DM though. If they for some reason rule the former though that means that Warforged are ill-suited to be monks or barbarians. It would be unlike them to shaft a race like that.

As @V2Blast points out Keith Baker the original author of Eberron goes into some intent on this page. However, once WotC declared him the winner of the world-building contest he no longer owns it and technically is not official ruling on the matter.

Moreover

Recent publications seem to have reverted the Warforged all the way back to what I believe was the original intent which was just a +1 to AC and donning and doffing armor albeit slightly differently than flesh and blood characters. Per Eberron -Rising from the Last War p36:

  • You gain a +l bonus to Armor Class.
  • You can don only armor with which you have proficiency. To don armor, you must incorporate it into your body over the course of 1 hour, during which you remain in contact with the armor. To doff armor, you must spend 1 hour removing it. You can rest while donning or doffing armor in this way.
  • While you live, your armor can't be removed from you

It does not appear that the Darkwood Core, Composite/Heavy plating made it to the final cut though. So this would be up to your DM to make a ruling.

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  • \$\begingroup\$ Comments are not for extended discussion; this conversation has been moved to chat. \$\endgroup\$ – mxyzplk - SE stop being evil Aug 25 '18 at 23:27
  • \$\begingroup\$ Note: the final version of the Integrated Protection trait in Eberron: Rising from the Last War and in the updated version of Wayfinder's Guide to Eberron has totally changed how it works, changing the answer to this question. \$\endgroup\$ – V2Blast Nov 23 '19 at 1:13
  • \$\begingroup\$ @V2Blast Thanks, looks like they went back to what I think the original was from way back, a bit disappointing but... oh well. \$\endgroup\$ – Slagmoth Nov 24 '19 at 1:48

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