It boils down to two main issues.
1. Does the compressor control circuit use 120V that it is tapping from one line to neutral inside of the compressor controls? If so, this won't work at all. If the power goes ONLY to the motor, then it would not matter to the motor if one side is grounded, all that matters is the potential difference between the two leads. In fact this is exactly what 230V single phase power looks like in most other parts of the world outside of North America, and motors are exchanged back and forth all the time. So if its only the motor, read on.
2. Is your 208V compressor motor OK with 223V? It might be, but it might not, only the compressor mfr can tell you for sure. If its a hermetically sealed refrigeration compressor, likely not, if it's an air compressor with a standard off-the-shelf single phase motor 5HP or under, it likely is, or read the actual motor nameplate if you can see it. If the nameplate says 208/230V, you're good to go. If not, someone put in the wrong transformer.