90 degrees phase shifted magnetic fields
The flux that is generated by a coil is

$$\Phi(t) = L\cdot i(t)$$

With L the self-inductance. If you now state that the transformer is ideal, then you also say that the flux through all coils is the same. You find that

$$\Phi(t) = L_{p1}\cdot i_{p1}(t) = L_{p2}\cdot i_{p2}(t)$$

This equation also implies that you cannot have a 90 degrees phase shift between the currents through the primary coils. You need

$$\frac{i_{p1}(t)}{i_{p2}(t)} = \frac{L_{p2}}{L_{p1}} = constant$$

You can basically treat it like someone saying:
I have two current sources in series, what's the total current?

To which the obvious answer is: It's impossible... Unless perhaps the currents are equal.