Closest rule to a case like this we have is the description of Constitution:
Temporary Bonuses: Temporary increases to your Constitution score give you a bonus on your Fortitude saving throws. In addition, multiply your total Hit Dice by this bonus and add that amount to your current and total hit points. When the bonus ends, remove this total from your current and total hit points.
Ability Damage: Damage to your Constitution score causes you to take penalties on your Fortitude saving throws. In addition, multiply your total Hit Dice by the Ability Damage penalty and subtract that amount from your current and total hit points. Lost hit points are restored when the damage to your Constitution is healed.
While there's no technical bonus or ability damage involved, this does show that it's intended for your hit point, both current and total, to shift instantly upon changes to your Constitution score.
Another comparison would be Possession:
The host’s soul is imprisoned with you, but can still use its own senses (though it can’t assert any influence or use even purely mental abilities). You can communicate telepathically with the host as if you shared a common language, but only with your consent. You keep your Intelligence, Wisdom, Charisma, level, class, base attack bonus, base save bonuses, alignment, and mental abilities. The body retains its Strength, Dexterity, Constitution, hit points, natural abilities, and automatic abilities. A body with extra limbs doesn’t allow you to make more attacks (or more advantageous two-weapon attacks) than normal. You can’t activate the body’s extraordinary or supernatural abilities, nor can you cast any of its spells or spell-like abilities.
In both the Possession and Synthesist cases, you flavorwise get the creature's body. Both note you use the body's physical stats, so similar treatment of hit points is only logical.
You can look at this from two points of view: Actual Hit Points or Effective Hit Points.
Actual Hit Points
Taking the above in account, and lacking any specific handling on the Synthesist page, your (non-temporary) hit points would decrease while your eidolon is "summoned".
This might be important for effects based directly on your hit point total, like Power Word Kill, or when non-lethal damage staggers you. However, these effects are generally quite rare.
However, due to having the Temporary Hit Points, as soon as those are depleted, your characters' Constitution is reapplied, instantly moving your HP total back up.
Which leads us to...
Effective Hit Points
Which I'll define as the amount of damage you need to take to get to 0 HP. This means that, using your calculation above, when having your eidolon summoned, your Hit Points are 10, your temporary Hit Points on top of that is 6, but your effective hit points would be 18.
Even if you take 18 damage in one hit, that would arguably take you to -2, but your eidolon would be gone, and with it your own Con score would apply again, and the rules cited above indicate you should immediately raise "current and max" hit points by 2, putting you at 0.