The phrase is an idiom in locative absolute. Sati is of course “to be”, not “mindfulness”. Similar idioms (tasmiṁ sati, evaṁ sante) mean something like “in that case”, “this being so”, etc.
Normally in Pali a repeated phrase is distributive. The point of the repeated phrase is to emphasize that it applies everywhere, in every situation. And the phrase does, in fact, usually occur in the same sentence as other phrases having this sense (yassa yassa … tatra tatreva). However most of these renderings—which I believe are based on the commentary—are limiting; they take the phrase as reducing the scope of application.
Normally—or maybe everywhere—the phrase occurs at the end of a section on meditation, when one has developed all the requisite qualities; it is then the culminating phrase. It has always felt odd to me why we should reach the end of a passage exalting the power and might of meditation, only to add a caveat saying “conditions apply”.
In AN 5.68, for example, the Buddha uses it when speaking of his own development of meditation. It culminates the development of the iddhipada, and seems to mean “having developed these things, I was able to get all the following good stuff, i.e. the various psychic powers”. It is hard to see how a limiting sense might apply here.
Further, the notion that āyatana means "base’ is also commentarial; it rarely if ever has that sense in the EBTs. Rather, the sense of āyatana in the EBTs is “field, opening, dimension”. It has the basic meaning of something stretched out, a field or dimension of activity within which something can occur. So I agree with Thanissaro’s sense for this.
So rather than limiting the application, I take it as universalizing or distributing it.
Yassa yassa ca abhiññāsacchikaraṇīyassa dhammassa cittaṃ abhininnāmeti abhiññāsacchikiriyāya, tatra tatreva sakkhibhabbataṃ pāpuṇāti sati satiāyatane.
Whatever things realizable by direct knowledge that they extend the mind to for the sake of realizing by direct knowledge, they become capable of realizing them, in each and every case.
Or more idiomatically:
They become capable of realizing anything that can be realized by insight to which they extend the mind, in each and every case.
