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The Summoner's Eidolon has a bonus to armour class based on level. The description of the Armor Bonus ability states this can be a mixture of regular armour bonus and natural armour bonus, as so:

The number noted here is the eidolon’s base total armor bonus. This bonus may be split between an armor bonus and a natural armor bonus, as decided by the summoner. This number is modified by the eidolon’s base form and some options available through its evolution pool. An eidolon cannot wear armor of any kind, as the armor interferes with the summoner’s connection to the eidolon.

So the Eidolon has an armour bonus, and the summoner decides how much of that is armour and how much is natural armour. However, it seems to me that there is no reason you would ever want to use any of this bonus as armour rather than natural armour.

Methods for improving natural armour bonus usually stack with existing natural armour; methods for gaining armour bonus do not - Mage Armour does not stack with an existing armour bonus, the Eidolon cannot wear real armour to be enchanted with an enhancement bonus, etc. Bonus used as armour is effectively wasted while the Eidolon is under the effect of Mage Armour (and when should it not be?) but natural armour is always useful.

Further, I am not aware of any difference in the way natural armour and regular armour functions which would make it desirable to have armour instead of natural armour. In fact, natural armour is sometimes at an advantage, i.e. Brilliant Energy weapons ignore armour but not natural armour.

Is there ever any situation where it would be advantageous for some of an Eidolon's armour bonus to be armour instead of natural armour?

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It is considered to be a fluff choice.

Natural armor and regular armor are considered equivalent equivalent in terms of balance. Even if there are a few differences between them (like brilliant energy weapons) designers don't seem to consider that's enough to balance them.

Considering splitting them, I guess it is possible to make you able to fluff your eidolon the way you want. If I want it to look like an armored bear the "logical" way to do that is putting a few points in natural armor (like what a bear get) and a lot of them in regular armor.

You are right about only putting points in one of them being more optimal, but for the sake of completion here is one advantage you could get from splitting them. Having at least +1 from armor makes you immune to damage from a whip, when it takes +3 from natural armor to be immune. So let's say you know mage's armor and you have only +2 to distribute, you could:

  • put everything in natural armor, for a total of +6 with mage's armor, but you are only immune to whips while the spell is working

  • put everything in regular armor, for a total of +4 with mage's armor. You are always immune to whips

  • split your points, for a total of +5 with mage's armor. You are always immune to whips.

If you are expecting to face a large amount of opponents with whips, the latter is the best, although that's not a really frequent situation.

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    \$\begingroup\$ If you are expecting to face a large amount of opponents who might use whips, I’d argue that the first one is best if it can convince them to use the whips. Whips are terrible, you’ll be weakening your opponents. \$\endgroup\$ – KRyan Apr 10 '17 at 12:51
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    \$\begingroup\$ I'm impressed that you actually managed to find a case where it'd be useful. As far as I can tell it only possibly applies when you have a single level of Summoner (since you get a +2 bump to armour bonus at 2nd, and all the base forms have +2 or better natural armour). But nonetheless, for a Summoner 1 in Whiptopia who doesn't know Mage Armour, it might be mechanically advantageous... \$\endgroup\$ – Carcer Apr 10 '17 at 14:28
  • \$\begingroup\$ Also: Spells and feats that affect either natural armor or worn armor. \$\endgroup\$ – ShadowKras Apr 10 '17 at 16:33
  • \$\begingroup\$ @ShadowKras Can you give an example of any feats/spells that would be relevant? \$\endgroup\$ – Carcer Apr 10 '17 at 21:37
  • \$\begingroup\$ Improved Natural Armor (B1) and Barkskin for natural. Blood Armor spell for regular armor. \$\endgroup\$ – ShadowKras Apr 10 '17 at 23:08
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You appear to be mostly correct - the best choice is to make the bonus either all armor or all natural depending on how you intend to improve the AC of the eidolon, with all natural being the better default choice.

In reality, though, there is very little practical difference between either choice. Most effects that ignore armor also ignore natural armor, with Brilliant Energy probably the only exception.

If the Eidolon could wear armor in some way, all natural armor bonus would be an even better default option.

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1st level spell: Mage armor!
2nd level spell: Barkskin is enchanment bonus to natural armor!

So: Put all “eidolon split armor” into natural armor bonus (untyped), that stacks with its +2 natural armor (untyped from its base form).

Total: +4 armor +4 shield + scaling natural armor from 3 stacable scources through all levels till 20!

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    \$\begingroup\$ Could you elaborate on your answer a little? Have you tried this yourself? \$\endgroup\$ – Jason_c_o Aug 8 '18 at 20:05

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