Let's say a summoner's eidolon is currently suffering from persistent bleed damage, is frightened 1, benefitting from Haste for 5 more rounds, and permanently blinded. The summoner is reduced to zero hit points, unmanifesting the eidolon (or perhaps the summoner spends the actions to unmanifest their eidolon). On the summoner's next turn, they re-manifest their eidolon (they received healing and are now conscious again). Does the eidolon come back with any of their conditions? Why or why not?
Conditions likely persist, with time passing as normal
Note: this is my summary of my player's research into the subject, with some additions from myself
We have to dig very deep to find any rules discussion on this: the best example I've seen is from the spell Summoner's Precaution. That spell provides a reaction to unmanifest your eidolon, with the following line:
Your eidolon unmanifests, and you can't Manifest your Eidolon for 1 minute. In exchange, you don't take the triggering damage, though your eidolon still suffers any other adverse effects that accompanied the damage.
This seems to imply that eidolons, even when unmanifested, can be afflicted by conditions (presumably the most common of the adverse effects that accompany damage). While it doesn't discuss what happens to the duration of those conditions (whether they are suspended or pass normally), since suspended conditions would be an exception to the normal condition rules, if conditions would be suspended we'd expect that to be explicitly stated. So, in the question's specific example, the eidolon would come back with 4 rounds of Haste, permanently blinded, still bleeding, and frightened 1, if the summoner hasn't had a turn yet (i.e. the eidolon was unmanifested when the summoner was reduced to 0 hp). If the summoner unmanifested the eidolon manually on previous turn, then they would have been able to reduce the frightened condition to 0, and potentially end the bleeding with a flat check.