Further perusal of the manual indicates that the main process output can provide PID control while the alarm output (or outputs, depending on brand/model) can simultaneously provide on/off control of a second process. You would want PID control for the dominant process (heat or cool).
Here's a circuit which should work with your Peltier device, using a MOSFET H-bridge to switch the polarity of a 12v (assumed) supply. The diodes ensure that all four MOSFETS don't conduct simultaneously if, for some reason, the heat and cool commands coincide; but you should be able to set the alarm and hysteresis parameters of the controller so that this doesn't happen.
Edit: Circuit and sim file updated. Reduced possible shoot-through pulse width.
Here's a circuit which should work with your Peltier device, using a MOSFET H-bridge to switch the polarity of a 12v (assumed) supply. The diodes ensure that all four MOSFETS don't conduct simultaneously if, for some reason, the heat and cool commands coincide; but you should be able to set the alarm and hysteresis parameters of the controller so that this doesn't happen.
Edit: Circuit and sim file updated. Reduced possible shoot-through pulse width.
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