Genso-eko (Virtue as instructed by Amida for retiring from the Pure Land to this world) is one of the important Jodo Sect doctrines (teachings of the Pure Land). It is one of the two aspects of Eko (Amida Tathagata's directing of virtue). Including Oso-eko (Virtue as instructed by Amida for going to the Pure Land), the alternative of the two types, both Eko together are called 'O-Kan Ni-eko' forming the primary doctrine of Jodo Shinshu (the True Pure Land Sect of Buddhism).
T'an-luan (early Chinese Pure Land philosopher and popularizer) explains Genso-eko in his main literary work "Muryoju-kyo Ubadaisha Ganshoge-chu" (Commentary on [Vasubandhu's] Upadesa on the Sutra of Immeasurable Life) (also known as Ojoron-chu or Wangsheng lun zhu - Notes on the Treatise about Birth (in the Pure Land)) as follows:
Instructions on virtue for returning to this world teaches that after being born in the pure land and completing the requirements for samatha and vipasyana, attaining the power of compassion, one can return and enter the deep forests of birth-and-death, teach and guide sentient beings, and bring together all who enter the Buddha-wayr.
He continues to explain the two types of eko as follows. Whether one is going forth or returning, it is all entirely for the sake of leading sentient beings across the sea of birth-and-death.
Therefore, 'It is to fulfill a compassionate mind by accepting virtue as first priority.'
Genso-eko means that one is endowed by Amida Tathagata with the ability to attain birth in the Pure Land of Amida Tathagata; to fulfill the practice of concentration and observation; to gain the power of enlightenment; to return to the world of birth-and-death, i.e. this world; to convert all sentient beings; and guide them to follow the way of Buddha together.
What should be noted here is that if Genso-eko is a being who attained birth in the Pure Land is bestowed by Amida with the ability to return to the defiled land taking the form of a bodhisattva to save people.
Interpreted as that, one is inclined to conceptualize something like a spirit which has returned from the Pure Land to this world. In fact some scholars do interpret it that way.
However, as indicated in a very forthright statement made by Shoma, a well known myokonin (a religious person): Some have happily picked up Namu-amida-butsu which I have happily spit from my mouth,' Genso-eko can be interpreted as hearing someone chanting the Name of the Buddha (Namu-amida-butsu), the sentient beings grasp that the existence that someone chanting the Name is the works of Amida Tathagata, and that the Name they themselves chant is the very form of returning from the Pure Land. This is the primary a concept of Genso-eko in the teachings of Jodo sect.
Also Shinran explains in "A Collection of Passages Revealing the True Teaching, the Practice and Realization of the Pure Land Way" as follows:
Benefiting from returning is an example of one's true intention of benefiting others. In this regard, Vasubandhu, the author, says the single broad unfettered mind awakens universally. Amida's compassionate instructions of virtue necessary for reaching the Pure Land and then returning again to this world, is made clear by the MasterT'an-luan; and with concern and care, he teaches the significance of those who are benefited and then go on to benefit others. Reverently embrace these words; receive them in deepest homage. As above, Genso-eko should be interpreted as ritagyo (altruistic practices).
Shinran further explains in "Passages on the Pure Land Way" as follows:
Second, Genso-eko is benefit of Ritakyoke-chi (competency to enlighten others to head for the True Pure Land). It came out from the 22th Vow of the 48 Vows. Also I named it as issho fusho no gan (Vow of succession to Buddhahood after one lifetime) or Genso-eko no gan (Vow of Genso-eko).
As above, he points the 22 Vow from the 48 Vows of Amida as the basis for his explanation.