Then the flux in the single core, must produce similiar results in both top and bottom halves of a single winding, even if the winding has three terminals on it.
It is no different than the results you would get from the top and bottom "halves" made from the flux in separate cores, or even the joining of the separate cores. The relationship at any instant in time is not an indicator that one direction across both windings is positive and the other direction is negative.
The single flux does not tell us that the direction we take the voltage for both halves must be the same. What about the "laws" tells you that one direction is "true" and the other direction is a math trick? The truth is, there is no law telling you that they must be taken the same, and if you think so I say post it and prove it.
With separate cores, we can prove that the positive wave of the "up" half is in phase with the negative wave of the "down" half. We can also use the positive forces from either end of the winding and show that these two positive forces have a 180? displacement. Polarity is a relative relationship and there is no universally "correct" direction.
The simple fact is that there is a positive pressure in both directions because the polarity is changing every 1/2 cycle. That is not the same as the battery case that keeps getting cited because the polarity never changes for the batteries.