Your circuit is 24VAC -> bridge rectifier -> 8Ω relay coil for around 3A. What kind of diodes? Silicon, selenium etc.?
It kinda looks like your rectifier is pooched, shorted or damaged diodes

With no load (and no filter cap), the rectifier diodes' leakage current can cause ghost AC voltage to be present.
Are you using LowZ to measure that? Oh wait 786 does not have that feature maybe. Or use a 10k resistor load at the bridge output.
Pulsing DC is not AC.
I don't know if the BM786 is AC-coupled or DC-coupled on ACV function. Fluke 87 and others are AC-coupled (have a series blocking capacitor) so it ignores any DC component that could be present. Other multimeters are DC-coupled, which can saturate the front-end and give bogus AC readings.
I think AC A is DC-coupled. You have to know your multimeter. I hope this is not too confusing.
Usually these circuits will have RF filtering a small film cap say 0.22uF to moderate voltage spikes and protect the rectifier diodes and lessen mains hash/EMI that can end up close to antenna circuits.