Driving a 12VDC solenoid from a 24VAC signal
The basics are simple:

Full wave bridge to make DC from the 24 V AC.  Assuming sine in, that means 33.9 V peaks, which results in about 32.5 V peak after two diode drops.

Significant energy storage after the full wave bridge.  This means a fairly large electrolytic cap.  This isn't for the normal purpose of reducing ripple, but to hold enough energy to pulse the solenoid after power is removed.

A small microcontroller with A/D that watches the rectified DC voltage.

A H-bridge like arrangement so that the solenoid can be driven with both polarities.  Polarity switching doesn't need to be fast.



When the micro first powers up, it knows that 24 VAC was just newly applied.  It waits until the storage cap is charged above some minimum limit, then pulses the solenoid with positive polarity.

The micro then continues to watch the rectified DC voltage.  When this drops below some threshold, it indicates that the AC input voltage has been removed.  The cap has to be large enough to still hold enough energy at that voltage to do a full solenoid pulse.  When the DC voltage drops to this threshold, the micro does a negative pulse.

After the negative pulse, the micro continues to watch the DC voltage.  If it goes above the "fully charged" threshold again, it assumes AC voltage was re-applied before the power supply dropped to the point that the micro no longer runs.  Otherwise, the micro dies of low power after the negative pulse, and does a positive pulse next time it runs again.

A reset supervisor that holds the micro in reset below some supply voltage would be a good idea.

There are different ways to deal with the solenoid being only rated for 12 V while the available energy is stored as 30 V or so:

Just do it anyway.  You are doing infrequent pulses of short duration.  Keep the energy dumped into the solenoid finite and low enough, and it won't get damaged.  However, the mechanism may not have time to react, or it may bounce with the short but forceful pulse, so see below.

Use a buck switcher with a enable input.  This takes the variable voltage on the capacitor and makes well enough regulated 12 V.  The micro controls the enable to do each pulse.

Run the solenoid directly from the capacitor, but PWM the low side switches of the H bridge.  A fixed PWM period may be good enough.  For extra credit, have the micro read the instantaneous DC voltage and adjust the PWM duty cycle accordingly each PWM period.  The solenoid is basically a inductor and will average the PWM pulse quite well.  It will "see" the DC voltage times the duty cycle, which you arrange to be about 12 V.