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So unlike 3.5, if you wear gear at least 24 hours, it counts as "permanent", qualifying you for things like feats, with the caveat that if you drop below, you lose access to the feat until you regain the prereqs. Does the same apply to Eidolons? They're only around as long as the summoner is awake, but they have their own gear - when dismissed it goes with them.
My goal is to give my Eidolons int boosting items to qualify for Combat Expertise and Improved Trip, Greater Improved Trip, and Fury's Fall, then make a quadruped cerberus with 3 tripping bite heads. The two size evos plus Enlarge Person results in a gargantuan Eidolon. It would have a Trip of 15 (bab) +3 (size) +15 (40 str) +3 (16 dex) +4 (feats) +5 (Amulet of Mighty Fists) +6 (belt of physical perfection) = 51, without buffs. 3 attacks (+1 with haste) with an attack bonus of 35. To compare to a trip unfriendly opponent, a CR 19 Ancient Red Dragon has an AC of 38 and a CMD of 56 against trip. 4 full BAB attacks on a pouncing charge with Haste, 85% chance of hitting each one, then each gets a free trip attempt with a 75% chance of success, resulting in an overall 98% chance to trip the dragon in a given round... which results in an AOO for another bite, and again when it stands up. If you get a druid friend with Strong Jaw (or UMD), each of those does 12d6+54, 5 hits with 85% hit chance averaging about 400 damage a round, more than the dragon's 362.
Note I would not actually bring this to a table - I am not that cruel to the GM or other players, I am leaving my PFS Eidolon medium. I do want to know if I can make it a medium creature tripping machine though :)
I would say, expect table variance. The eidolon isn't "defined" when it isn't around and the only thing we have to rely on is flavor text in this regard.
RAW, it is around when the summoner calls it. The item effect you are looking for takes 24 hours to gain. So if you manage to keep it summoned for the 24 hours you are golden. The moment the eidolon "ceases" is where you can expect the variance. You can flavor it as them lounging around in another dimension, but you can aso flavor it as not existing at all until summoned.
If you want to pursue the build, look into things that will remove your need to sleep or the like. That way you have what you want "effectively" for the period you are adventuring. For PFS, I would error on the side of caution, just because you may run into a GM who won't let do it.