theoretically, you might both even have exactly the same evidence, but gathered in a different order. The question is one of differing interpretations, not raw data as such.
I happen to be studying conflicts in a completely different domain, in which I claim the solution is to ensure the events shape identical result no matter in which order they are applied. I briefly wondered whether my result could be useful in other domains, and I thought of lesswrong: perhaps we should advocate update strategies which don’t depend on the order in which the evidence is encountered.
And then your post came up! Nice timing.
The reason we shouldn’t update on the “room color” evidence has nothing to do with the fact that it constitutes anthropic evidence. The reason we shouldn’t update is that we’re told, albeit indirectly, that we shouldn’t update (because if we do then some of our copies will update differently and we will be penalized for our disagreement).
In the real world, there is no incentive for all the copies of ourselves in all universes to agree, so it’s all right to update on anthropic evidence.