Two-core transformer with three windings?
Many old literature can be found. This is a magnetic amplifier, those days, the power amplifiers were made similarly as that. Book



EDIT:

If the magnetic cores are isolated and there is no common magnetic path, then the result is much simpler:

$$u_i\propto\dfrac{d\Phi}{dt}$$
$$u_{sec}\propto\dfrac{d\Phi_{prim}}{dt}+\dfrac{d\Phi_{ctrl}}{dt}$$

Both fluxes (primary and control) are due to the magnetising currents in primary and control winding. The secondary winding "see" their sum, so the secondary voltage is the sum of both primary voltages. Same as you would have two separate transformers with secondaries wired in series.