This is where we totally disagree, since our answers to that question will differ. Relative to the 'neutral' (centre tap), the currents through R1 and R2 are 180° out of phase. When current is travelling towards the neutral through R1, then current will be travelling away from neutral through R2. Relative to neutral, that is a phase difference of 180°.
But that's no different than the situation which exists with the simple 2-wire connection before we added the neutral, and you've already agreed that such cannot be anything but single phase.
Go back to that basic 2-wire connection and examine the currents at the midpoint between R1 & R2. Current flowing
toward that point from R1 is flowing
away from that point into R2 (it couldn't do anything else, obviously, as there's nowhere else for it to go - Kirchoff's First Law at its most basic). Adding the neutral doesn't change that, it just provides an alternate path for the current to flow if R1 & R2 are unequal.
Look at it in terms of Kirchoff's Law if you want. For simplicity, consider the situation in which R1=R2, hence no current through the 'neutral'. Per K's Law, the currents at the point where the resistors join must sum to zero. hence, if (at any point in the cycle), the current through R1 (relative to neutral) is X amps, then the current through R2 (relative to neutral) must be -X amps. If the current through R2 (relative to neutral) is always -1 times the current through R1, that is, by definition, a phase difference (relative to neutral) of 180°.
Again, if you want to consider a current flowing toward any given point as X amps and current flowing away from that point as -X amps, the same applies to the basic 2-wire circuit. It has to apply to
any arbitrary point you care to select in
any simple series circuit.
As you say, if R1 & R2 are equal, adding the neutral changes nothing. The instantaneous value and direction of current in both R1 & R2 at any given moment are exactly the same as they would be without the neutral, so with absolutely no change in the currents, how can a single-phase system have become a 2-phase system?
Your argument would only 'work' if one measured phases relative to something other than 'neutral' (e.g. one of the 'live' supply conductors), but I can't see much sense in doing that.
I think that's where the problem lies; you seem to want to look at everything (relative voltages and currents) with respect to the neutral as soon as the latter is introduced. Why? There's nothing special about it in respect to the number of phases; it's just there to carry any imbalance current if the loads on each pole or phase are unequal. How would you regard a 3-phase delta arrangement where you don't have a neutral?
I'm not sure what you mean by "change the phase relationships" since, as we have agreed, when there are only two wires (and no other reference), phase is meaningless. However, again as above, when one adds a neutral, the currents through R1 and R2 relative to that neutral will be 180° out of phase.
I mean the phase of the current through R1 relative to the phase of the current through R2. In the simple 2-wire connection, current starting at zero in both will increase sinusoidally in both until it reaches its peak in both at the same instant, then it will decrease and pass through zero again at the same instant, reach its peak in the opposite direction at the same instant, and so on. The current through the two resistances is in phase. It can't not be (disregarding tiny amounts of stray capacitance and inductance which might introduce a tiny phase shift, but obviously we've been assuming that all along).
Adding the neutral connection in the 3-wire system doesn't alter that in any way: The instantaneous current will still be zero in both R1 & R2 at the same moment, still reach its peak at the same instant, etc. In other words, adding the neutral results in absolutely no change in the phase relationship between the currents in R1 & R2. In the balanced load situation where R1 & R2 are equal, connecting and disconnecting the neutral has absolutely no effect on anything, even the value of currents, so how can connecting it create a second phase?
If you do the same thing with 2 phases of a 3-phase wye supply, you
will see a phase shift in the currents through R1 & R2 when you add the neutral connection to their mid-point.