The FTL phone argument is not a single argument, but a strategy to counter different arguments used to defend relativistic causality, like "relativistic explanations" of the violations of Bell's inequality. In general, it works in the following way:
Alice: (Some argument which seems to show that the violation of Bell's inequality and Einstein causality are compatible.)
Ilja: Assume we have build a black box device which works like an FTL phone. We can talk with a station on Mars without any observable time delay. Then, Einstein causality is obviously wrong, falsified by observation of this phone. Do you agree?
Ilja: Fine. Now let's try to apply your argumentation to our FTL phone. Hm. It seems, we need only some minor modifications of your argument to apply it to the phone: (Some variant of the argument. It seems to show that the FTL phone and Einstein causality are compatible.)
Alice: Hm. But this argumentation is nonsense. Its clear that the FTL phone violates Einstein causality.
Ilja: Indeed. But why this variant of the argument is nonsense, but your original argument not?
Alice: ..., hm, ...
Ilja: Indeed. I'm very interested in your modification.
The FTL phone argument is not universal. It is even not intended to meet all arguments. The main purpose is to subdivide arguments into two subclasses: really important arguments which are based on really important differences between the EPRB device and an FTL phone, and arguments based on confusion.
One example where the FTL phone argument cannot be applied is the popular argument that violations of Bell's inequality cannot be used to construct an FTL phone.
Price (arXiv:gr-qc/9406028, abstract): Bell's theorem depends on the assumption that hidden variables are not influenced by future measurement settings. ... The suggestion [that this assumption is invalid] has generally been regarded as outlandish ... This paper argues that thereis surprisingly little justification for this reaction.
(p.10) ... Hidden variables are thus dependent on the fate of the agents concerned ... Unlike ... instantaneous action-at-a-distance, this advanced action proposal does not conflict with special relativity. This is because the point at which the twins become coupled ... lies within the light-cones of both their later interrogations. The effect is not instantaneous and not at a space-like distance.
Of course, if we introduce an influence backward in time, from future measurement settings, we can explain, in a similar way, an FTL phone too. At least I cannot see any difference.
Alice: Correlation is not causation. All what we observe are (quite weird, I agree) correlations between the decisions of the experimenters and the results of the experiments.
But how do we establish that these weird correlations really exist? We establish this only later, by comparison, and we use for this purpose classical, non-FTL channels.
Ilja: Correlation is not causation for the phone too. All what we observe are (quite weird without assuming FTL, I agree) correlations between the decisions of the experimenters (some words) on one side and the results of the experiments (the sound waves we hear on the other side).
But how do we establish that these weird correlations really exist? We establish this only later, by comparison, and we use for this purpose other channels.
Notes: Indeed, what else would prove that the phone is not a fake? Assume there are some sound bytes which Alice is able to create and Bob would accept as a proof once he hears them. Why should a deceiver on Bobs side be unable to create similar information to fake Alices message? Only to compare Alices input and Bobs observation later, using classical, reliable information transfer channels, gives us certainty that the phone is not a fake.
Bell (Speakable and unspeakable in quantum mechanics, Cambridge 1993, p.154): ... it may be that it is not permissible to regard the experimental settings a and b ... as independent variables, as we did. ... Now even if we arranged that a and b are generated by apparently random radioactive devices, housed in separate boxes and thickly shielded, or by Swiss national lottery machines, or by elaborate computer programs, or by apparently free willed experimental physicists, or by combinations of all of these, we cannot be sure that a and b are not significantly influenced by the same factors l that influence A and B ... Apparently separate parts of the world would be deeply and conspiratorically entangled, and our free will would be entangled with them.
This argument may be applied immediately to the FTL phone. No necessity to change even a single word.
Alice: We have a theory which describes this: QFT. And in this theory we have no FTL effects.
Ilja: The world where the FTL phone exists is a hypothetical one. Let's specify this hypothetical world in a little bit more detail.
Let's start with a description how the FTL phone in this world really works. We assume that there exists some complex field P(x,t) which follows the classical Laplace equation as in Newtonian theory for the gravitational potential. Therefore, it distributes information over the whole universe immediately. But it interacts with usual visible matter only in some weak way. All we can observe via the weak interactions is some probability distribution which equals |P^2(x,t)|, but not P(x,t) itself.
Scientists in our hypothetical world are influenced by positivistic argumentation, and because they cannot measure P(x,t) they deny the reality of P(x,t). They accept P(x,t) (as well as the Laplace equation for it) only as a mathematical tool which allows to compute the probability distributions |P^2(x,t)|. (Similarities with existing theories are not really accidental.)
Based on this theory these scientist can, as well, make similar claims. They have a theory which describes the phone. And this theory does not make any claims about what really happens. It restricts itself to the prediction of probabilities. These probabilities it predicts successfully.
As long as we do not look at the mathematical differences between the two theories, the FTL phone argument works. More interesting is that it works also for some arguments supported with mathematics.
For example, attempts to prove that the FTL phone is really a causal influence may be based on properties of the P-theory as described here, for example the non-local character of the equation for P. Such attempts probably do not survive the FTL phone test because similar arguments based on pilot wave theory are possible too:
What does not take over to our P-theory is the locality theorem of QFT. Operators defined for space-like separated spacetime points commute. This makes it impossible to use QFT devices to send information FTL. Thus, we are back to the basic property which distinguishes EPRB devices and the FTL phone.
Typical verbal descriptions about how the MWI explains the correlation (which involve various splits of the universe) do not seem to be specific enough to distinguish between EPR and an FTL phone. Why not talk about FTL phones using "splitting universes" and name this "explanation"? You can be quite sure, given the absence of any equations in many worlds, together with the problems of many worlds to incorporate a reasonable notion of probability into the theory, that you will never meet a many worlds argument specific enough to distinguish a violation of Bell's inequality from an FTL phone.
Similar remarks. If we have an FTL phone we can also describe it using expressions containing "consistent histories". I have never seen a presentation of the argument which has been detailed enough to understand why the "consistent histories explanation" explains EPR without FTL but a similar talk about an FTL phone not.