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Here is a solution, or more precisely a family of solutions.

The white king is on and the last few moves were Proof: Let's consider these possibilities in order. Therefore Now, This is no good yet, because

Wrong answer (I made a boneheaded mistake, kindly pointed out in comments by @RShields, to which I have drawn attention below.)

In the position shown, Now, Before that Black White So we have the following sequence of events. Now, We still haven't figured out Suppose that Now So let's consider W …

Certainly not an optimal solution Here is a way to get to that position, as a baseline for others' improvements. I bet it can be reduced quite a lot: 1. b3 d6 2. Ba3 Nc6 3. Bc5 dxc5 4. Qc1 g5 5. Qa3 …

HTM's answer is very nice and must be what's intended here. But I'm not quite seeing why the puzzle isn't cooked as follows:

Grossly incomplete answer (I misread the question; having had this kindly pointed out in comments I spent a short while seeing whether I could finish the job before needing to go to bed tonight, but i …

Alice's last move must have been because Clearly Bob's previous move was Now consider We can't

This is an at-most-half-baked answer, but I begin to suspect that the trouble lies For a (ridiculous) example, This sort of possibility is maybe made more plausible by

I hope this is not the correct answer, but perhaps In which case,

This game is So in this case If we take "rational" to mean that when either player had a winning move they took it then Now So