You can use your manufactured weapon as normal for full attacks – iteratives, Two-Weapon Fighting, Flurry of Blows, whatever.
You then get any natural weapon attacks, all treated as secondary attacks (−5 penalty to attack, Strength to damage halved), unless the limb involved in the natural attack was already used for a manufactured weapon. This usually means no Claws, but other restrictions could imaginably come into play.
Worth noting that the Multiattack feat reduces the penalty on secondary attacks to −2 instead of −5. Can be worth it if you’ve got three or more.
"Multiattack" refers to two different rules items here. One is a class ability. The other is a feat.
As part of the Eidolon class feature, the Eidolon gains a special ability called "Multiattack" when the Summoner reaches 9th level. This is a special ability, not a feat.
The text of that special ability is:
An eidolon gains Multiattack as a bonus feat if it has 3 or more
natural attacks and does not already have that feat. If it does not
have the requisite 3 or more natural attacks (or it is reduced to less
than 3 attacks), the eidolon instead gains a second attack with one of
its natural weapons, albeit at a –5 penalty. If the eidolon later
gains 3 or more natural attacks, it loses this additional attack and
instead gains Multiattack.
The special ability grants a feat (also called "Multiattack") if the eidolon has 3 or more natural attacks. Otherwise, the special ability grants a second attack.
Can the Synthesist's eidolon gain that feat? No.
The Synthesist's "Fused Eidolon" special ability says this:
The eidolon has no skills or feats of its own.
Therefore at 9th level, when a Synthesist's eidolon gains the Multiattack special ability, the effect depends on the number of natural attacks that the eidolon has.
- If the eidolon has 3 or more natural attacks, the Multiattack special
ability has no effect for the Synthesist's eidolon, since the special
ability would usually grant a feat, while the Synthesist's eidolon
cannot have feats.
- If the eidolon has less than 3 natural attacks, the Multiattack
special ability grants a second attack, as proscribed in the ability text.
Does the Synthesis gain the benefit of this ability? Yes. If the eidolon gains a natural attack via this special ability, then the Synthesist, when fused with the eidolon, gains that attack. From the Fused Eidolon class feature text:
The synthesist also gains access to the eidolon’s special abilities
and the eidolon’s evolutions. The synthesist is still limited to the
eidolon’s maximum number of natural attacks.
Best Answer
They don't, but they don't really need to.
So your Natural Armor bonus will not stack with that of your Eidolon, since both bonuses are of the same type, and there is no exception to the general rule found for the Fused Eidolon ability.
However, look at the following parts
(emphasis mine) Note that you do not lose your Natural Armor while fused (because the rules don't say so)
So instead of taking your Eidolons Natural Armor Bonuses, set them to zero. Put all your Eidolons Armor Bonus to regular armor and keep your own Natural Armor.