So, for this issue I would note that for the coinflip to influence the decision algorithm, there needs to be an arrow from the coinflip to the decision algorithm. Consider two situations:
Omega explains the counterfactual mugging deal, learns whether you would pay if the coin comes up tails, and then tells you how the coin came up.
Omega tells you how the coin comes up, explains the counterfactual mugging deal, and then learns whether you would pay if the coin comes up tails.
Those have different Bayes nets and so it can be entirely consistent for TDT to output different strategies in each.
So, for this issue I would note that for the coinflip to influence the decision algorithm, there needs to be an arrow from the coinflip to the decision algorithm. Consider two situations:
Omega explains the counterfactual mugging deal, learns whether you would pay if the coin comes up tails, and then tells you how the coin came up.
Omega tells you how the coin comes up, explains the counterfactual mugging deal, and then learns whether you would pay if the coin comes up tails.
Those have different Bayes nets and so it can be entirely consistent for TDT to output different strategies in each.