If a Mythic Unchained Summoner's Eidolon is hit with the Imprison spell, does it automatically unsummon, or is it trapped in the prison?
An Eidolon will be affected as normal by Imprisonment.
Target creature touched
When you cast imprisonment and touch a creature, it is entombed in a state of suspended animation (see the temporal stasis spell) in a small sphere far beneath the surface of the ground...
...You place the subject into a state of suspended animation. For the creature, time ceases to flow, and its condition becomes fixed. The creature does not grow older. Its body functions virtually cease, and no force or effect can harm it. This state persists until the magic is removed (such as by a successful dispel magic spell or a freedom spell).
Nothing in the spell allows for Outsider creatures to be immune to it or avoid it.
This is also not the same as killing the Eidolon, so the Eidolon does not return to its home plane by this section of the Eidolon ability.
...except it is not sent back to its home plane until reduced to a number of negative hit points equal to or greater than its Constitution score.
The GM may or may not allow the Eidolon to be
dismiss[ed] by the summoner (a standard action).
... but in most cases they won't because of how powerful of a spell Imprisonment is and the small list of options allowed for overcoming it (to include that Wish and Miracle do not work). At best, as a GM, I'd allow you to dismiss and summon your Eidolon to/from their home plane, but they would still be affected as Temporal Stasis.
An eidolon that fails its saving throw against imprisonment is affected like any other creature…
Nothing in the description of the eidolon says that when an instantaneous effect affects the eidolon that the eidolon is automatically sent back whence it originally came. In fact, all sorts of tragedies can befall a hapless eidolon, but the eidolon vanishes instantly typically only if it's slain or affected by an effect like the spells banishment or dismissal or if the summoner dismisses it, sleeps, is rendered unconscious, or is himself slain. Since the spell imprisonment does none of those things to the summoner or the eidolon—instead, the eidolon's "entomb[ed] in a state of suspended animation in a small sphere far beneath the surface of the ground"—, the eidolon suffers the imprisonment spell's effects normally.
The imprisonment spell's description continues, saying
The subject remains there unless a freedom spell is cast at the locale where the imprisonment took place.… A wish or miracle spell will not free the recipient, but will reveal where it is entombed.…
…And this implies to this GM that nothing else but freedom spell will loose the eidolon from the sphere created by the imprisonment spell and return it to the battlefield, ready once more to engage in the current battle. (The freedom spell's casting time is just one standard action, after all!)
…But this GM would still allow a summoner to dismiss his imprisoned eidolon
This GM would allow a summoner to go to sleep, to be rendered unconscious, to have himself killed (likely a last resort!), or to take a standard action to dismiss his eidolon were the summoner's eidolon to have been the subject of the imprisonment spell. Then, when he awakens, recovers, is brought back from the dead, or finishes the dismissal process, the summoner can bring forth his eidolon normally with his 1-min. ritual.
The temporal stasis spell—referenced by the imprisonment spell—, in part, says, "For the [affected] creature, time ceases to flow, and its condition becomes fixed," but if that creature is the product of magic—like a summoned creature, which is what the eidolon is frequently likened to—that magic that brought it forth remains. Thus in the same way that a summoned creature that's the subject of an imprisonment spell would disappear from the imprisonment spell's sphere upon the summon spell's duration expiring, so, too, can an eidolon be dismissed while it's in the imprisonment spell's sphere. Dismissing the eidolon while it's trapped does not technically free the eidolon from the imprisonment spell's sphere. The eidolon doesn't immediately rejoin the battle in progress, and the actual prison sphere's still there; it's just that eidolon's been sent home, leaving the sphere it formerly inhabited empty. And, thereafter, the summoner can to bring forth the eidolon anew, typically by the summoner performing his 1-min. ritual.
While to some this may seem a weak, technical rules exploit that limits the imprisonment spell's efficacy, bear in mind that a successful imprisonment spell still usually removes the typical eidolon from the current battle—the usual goal for NPCs—and won't see a summoner—be the summoner PC or NPC—lose a valuable class feature until the summoner can return to the spot where the eidolon was the subject of the imprisonment spell with—somehow!—a freedom spell ready a cast—that spell, by the way, in addition to being a 9th-level spell, isn't on the summoner's spell list.
It should affect the eidolon normally, but you may summon a new eidolon
The design intent for the eidolon is to be part of the summoner, as a physical manifestation of a creature concept they have in mind and are able to bring to life using their magical powers. Means of permanently removing an eidolon from the game are frowned upon by the developers nearly every time they are asked about it, as seen in this example, by James Jacobs (Creative Director) when a user asked what would happen if the eidolon's soul was eaten during an encounter:
At their current level if the encounter go badly for the PC (improbable but possible) one of them will be killed with little chance of returning to life. The most logic target would be the summoner (partially because she is one of the top targets in the BEEG list, partially because with 6 players a summoner is a problem).
So I am considering the different possible effects of having the summoner (or her eidolon) soul permanently imprisoned.
1) the summoner is killed, the eidolon disappear and the player create a new character. I don't like it much as the character is well engrained in the campaign.
2) Giving a new ability to the life link power of the summoner, the eidolon sacrifice her existence to save the character. The player keep his character and after a period of mourning and soul searching get to re-make her, choosing a new class with retroactive effects (i.e. she would get change his class from level 1 onward). It would be the better option for continuity, but I am at loss about what class would be the best one for the character. Any suggestion? (naturally the player will have full choice, I only want a suggestion).
3) Same as above but the player maintain her class, without an eidolon. No good. A summoner is too strong but I don't think that removing a major feature of a class will be a good idea.
4) The PC die but the player can continue to play using the eidolon as an unfretted eidolon capable to gain character levels. Story wise it has possibilities but I don't know how balanced it would be.
Any other possible solution?
Now... the eidolon is a fragment of the summoner, and if it succumbs to the soul eating... it just dies. It can't be permanently removed from play in this manner since it's a physical representation (being an outsider) of a portion of the summoner's soul. So... if it's killed, it just goes away and can be brought back normally. It's only the summoner himself who can be soul drained. Having an eidolon doesn't put a character into double jeopardy.
That said, a GM that pulls a Imprisonment or Temporal Stasis against an eidolon, without an easy way to remove it as a short term solution, is removing half of what makes your character what it is from the game, and will leave a bad taste in that player's mouth.
It's also worth noting that an eidolon is not a specific creature, but an "aspect" of a creature created by the summoner (similar to Summon spells). That said, even if the current eidolon is imprisoned/killed/soul drained, that shouldn't prevent the summoner from summoning another aspect of that same creature the next time they use their Eidolon Summoning Ritual.
The eidolon forms a link with the summoner, who forever after summons an aspect of the same creature
While the rules are silent on what happens to conditions on your eidolon when it is dismissed, everything in the rules about eidolons seems to indicate they are just an improved (and customizable) Summon, and not the exact same creature every time they are summoned. For instance, an eidolon that was petrified can be dismissed and you may summon another aspect of your eidolon that should no longer have the petrified condition, as the only thing that remains the same between summons is their hit points:
A summoner can summon his eidolon in a ritual that takes 1 minute to perform. When summoned in this way, the eidolon’s hit points are unchanged from the last time it was summoned. The only exception to this is if the eidolon was slain, in which case it returns with half its normal hit points. The eidolon does not heal naturally. The eidolon remains until dismissed by the summoner (a standard action). If the eidolon is sent back to its home plane due to death, it cannot be summoned again until the following day. The eidolon cannot be sent back to its home plane by means of dispel magic, but spells such as dismissal and banishment function normally. If the summoner is unconscious, asleep, or killed, his eidolon is immediately banished.
Using James Jacobs words about eidolons ("a portion of the summoner's soul"), it seems their life force are connected to the summoner's life force, especially since they are also dismissed when the summoner is asleep or dies. That would also explain why their hit points are fixed between summons, rather than regenerating naturally or being replenished when you summon another creature. As such, I would say that every time you summon your eidolon, you are summoning a new creature, not the same as before.