If you're saying that you wired up a 220v version to 110v then I may be able to shed some light.
The internal transformer of the 110v version outputs a nominal 9v.
The internal transformer of the 220v version outputs a nominal 10v.
The relays for both versions are 12VDC with a pickup (must operate) voltage of 9v and a dropout (must release) voltage of 1.2v.
Relay operation between the pickup and dropout voltage can be inconsistent and should be avoided in design and operation.
If 110v is applied to the 220v transformer, approximately 5v will result. 5v is likely enough to power the integrated circuits and led's in the controller but may not be enough to confidently switch and hold the relays in position.
Bottom line? It might work. But, if the relays do not work consistently, and chatter, it may harm the compressor of the fridge.