AC relays do exist.
I have seen 120VAC relays.
They are not that common, but not that uncommon either.
Old-school Air Conditioning units use a 24VAC to switch the Compressor (and possibly also the fan) on.
You can just use a 24V transformer and a 24VAC relay.
Or you can just use 3 diodes, a resistor (to reduce the voltage a tad), and a DC relay.
You can get away with 1 diode, but 3 is better.
Example circuit:
Connect "AC Neutral" to the "negative" or "positive output" of a rectifier diode, and connect the other pin of the diode to one pin of the relay coil.
Connect "AC line" to the "positive input" of a second diode.
Connect a resistor (about 10Ω) between the unconnected pin of the second diode and the unconnected pin of the relay coil to reduce the voltage (the voltage of rectified DCreachs the AC voltage times the square root of 2). (24VAC rectifies to as high as 34VDC)
Connect a third diode facing "backwards" across the relay terminals to deal with the inductive reverse voltage released by the relay coil.
This could be done with just 1 diode and a resistor, but the 3 diode approach is probably safer (the diodes block half the AC) and more efficient (reverse voltage is shorted by the diode for free, instead of contributing to the power consumption of the device)