What does it mean that a first-bhūmi bodhisattva permanently subdues nature-obstructions like an arhat?

"Nature" refers to the fundamental nature of sentient beings. The fundamental nature of sentient beings is the mental nature of the manas, the mental factors (caittas) of the manas. Precisely because the mental nature of the manas is unwholesome, it has constantly been obstructing sentient beings from attaining unsurpassed bodhi. Therefore, only by eliminating the obstructions of the manas's mental nature can Buddhahood be accomplished.

Bodhisattvas at the first bhūmi begin to gradually eradicate all the afflictive habits of the manas, but it is impossible to eradicate them completely. When they cannot be completely eradicated, some of the afflictive obstructions (kleśāvaraṇa) that affect birth and death within the three realms must be subdued. From another perspective, if the manifest afflictions of the manas were completely eradicated, one would, like an arhat of the fourth fruit, completely eradicate attachment to the self of persons and would enter nirvāṇa without remainder, being unable to accomplish Buddhahood. The Buddha required bodhisattvas from the first to the seventh bhūmi not to enter the definitive stage of the path of liberation—that is, not to take up the fruit of the fourth-fruit arhat—lest they inadvertently enter nirvāṇa without remainder.

Therefore, bodhisattvas at the first bhūmi must retain a little bit of craving and attachment, so as to ensure that they can still be reborn in the world of the three realms. But they must not retain serious craving and attachment; they may only retain a slight, mild craving—having a bit of attachment to things like meditative concentration (samādhi) would suffice, and they may also have attachment to the Mahāyāna Dharma. That is, the craving of the manas is forever subdued rather than completely eradicated, so that life after life they can benefit themselves and benefit others, and even until they become Buddhas, they will not enter nirvāṇa without remainder.