They do have inflation adjusted annuities. ...

They're about as rare as unicorn fins : )
Probably some, but not easy to find! And probably very very expensive.
There are lots of annuities which have a fixed increase per year, say 2%, which are often misleadingly sold as inflation-adjusted.

FWIW, the last annuity quote I looked at (after the recent inflation and interest spike):
Joint life annuity for a couple both aged 75 today, payments deferred for ten years to get a higher monthly income. A "longevity risk" tool.
No "inflation" increase because in this case it cost so much that the odds are you'd be better off without it.
The result, assuming we see 2.5% inflation: they would not breakeven unless and until one of them reaches age 109.
That is, until that age the real money put in is still more than the real money back out.
Since it is somewhat unlikely that one will live to that age, it is likely a loss from an investment point of view.
You pay a lot for the certainty of an annuity...generally far more than what it's worth. (the "actuarially fair" price)

For comparison, consider simply withdrawing from a "deferred" TIPS ladder.
Put all the money into TIPS, assuming long tips yield is same as today's 1.755%. Assume 2.5% inflation again.
Same capital put in, the same 10 year deferral before first payment, same first payment in real terms as with the annuity above.
(but in this case the payments keep up with inflation thereafter rather than staying flat in nominal terms)
The money would last to age 112. And assuming they die earlier, there is capital left in the estate unlike with an annuity.
Thus the "cash out" (real balance in the bank plus real payments received to date) is positive at all dates, rather than starting at age 109.

Jim