Eternal black holes and Hawking radiation

I have a fairly simple question which is confusing me a lot.

As Hawking showed, a black hole originated by collapse will emit Hawking radiation. This process will reduce the mass of the black hole until it will evaporate. But what about black holes which are not originated by collapse, i.e. eternal black holes?

In case we have a black hole in AdS, the black hole will be indeed eternal because all the emitted Hawking radiation will eventually come back to the black hole, since AdS acts like a confining box.

But if we consider a eternal black hole in flat space there is no process that ensures the stability of the black hole. So it would mean that at a certain point it will evaporate. How is this black hole eternal? To put it in a different way, how can a geometry which is invariant under time translations identify a precise time, the one at which the black hole evaporates?