Nohow. Your decision procedure's output leads to money being put into one box, and also to you choosing that box, or little money being put into second box, and also to you choosing both.
If you ever anticipate some sort of prisoner dilemma between identical instances of your decision procedure (that's what Newcomb's problem is) you adjust the decision procedure accordingly. It doesn't matter in the slightest to the prisoner dilemmas whenever there is temporal separation between instances of decision procedure, or spatial separation; nothing changes if the Omega doesn't learn your decision directly, yet creates items inside boxes immediately before they are opened. Nothing even changes if the Omega hears your choice and then puts items into the boxes. In all of those cases, a run of decision procedure leads to an outcome.
prisoner dilemma between identical instances of your decision procedure (that's what Newcomb's problem is)
I'm not so sure. The output of your decision procedure is the same as the output of Omega's prediction procedure, but that doesn't tell you how algorithmically similar they are.
Occasionally a wrong idea still leads to the right outcome. We know that one-boxing on Newcomb's problem is the right thing to do. Timeless decision theory proposes to justify this action by saying: act as if you control all instances of your decision procedure, including the instance that Omega used to predict your behavior.
But it's simply not true that you control Omega's actions in the past. If Omega predicted that you will one-box and filled the boxes accordingly, that's because, at the time the prediction was made, you were already a person who would foreseeably one-box. One way to be such a person is to be a TDT agent. But another way is to be a quasi-CDT agent with a superstitious belief that greediness is punished and modesty is rewarded - so you one-box because two-boxing looks like it has the higher payoff!
That is an irrational belief, yet it still suffices to generate the better outcome. My thesis is that TDT is similarly based on an irrational premise. So what is actually going on? I now think that Newcomb's problem is simply an exceptional situation where there is an artificial incentive to employ something other than CDT, and that most such situations can be dealt with by being a CDT agent who can self-modify.
Eliezer's draft manuscript on TDT provides another example (page 20): a godlike entity - we could call it Alphabeta - demands that you choose according to "alphabetical decision theory", or face an evil outcome. In this case, the alternative to CDT that you are being encouraged to use is explicitly identified. In Newcomb's problem, no such specific demand is made, but the situation encourages you to make a particular decision - how you rationalize it doesn't matter.
We should fight the illusion that a TDT agent retrocausally controls Omega's choice. It doesn't. Omega's choice was controlled by the extrapolated dispositions of the TDT agent, as they were in the past. We don't need to replace CDT with TDT as our default decision theory, we just need to understand the exceptional situations in which it is expedient to replace CDT with something else. TDT will apply to some of those situations, but not all of them.