Relay Coil should get full 12V or 0V
In the schematic shown below, the entire secondary of the transformer has been used, two rectifiers have been eliminated, and C1 is used to smooth the full-wave rectified 16.5 volts. 

U1 is a 12 volt linear regulator which drops the 16.5 VDC down to 12VDC for the relay and whatever else the 12 volts is being used for, R6 representing both the relay coil and the remaining parallel load, if there is one.

U2 is a voltage comparator with an internal 400 millivolt reference. The voltage divider R3 R4 is used to set the voltage at the non-inverting input of U2 to slightly over 400mV when the output of U1 is at 12 volts. That will force the output of U2 high, turning Q1 ON and energizing the relay coil.

If, for some reason, U1 OUT should fall below 12 volts, the voltage at U2+ will fall below 400mV, U2's ouput will go low, and the relay will become de-energized, opening its normally-open contacts.

U2 is supplied with about 6.5 millivolts of internal hysteresis and if more is needed it can be applied externally as shown on page 10 of the data sheet.