Re: fuse or not
As Bennie stated, A/C Compressors are highly susceptable to extended Locked Rotor time periods, and flat-out failing to run out from Locked Rotor (Stalled / Unmovable Rotor).
OCPD at the equipment should be able to allow the high Overcurrent and Long Time frame of starting, but also prevent excessively high LRA and be dialed into the unit's rated FLA, which will offer running protection.
If fused properly (within close tolerances of the running load amperes), the problem of Single Phasing a Three Phase Motor can be covered.
Once again as Bennie stated, the Locked Rotor starting situation is similar to a Short Circuit, which will tend to blow out at least Two fuses (connection is L-L). If the fuses are sized too large, they may not react in time to an attempted 1? start (or fried rotor), and results in a loss of Smoke from the Motor's Windings.
Lose too much smoke and the flames may be released too!
If a Polyphase (3?) Induction Motor experiences a Single Phase situation while it's running, the lines which are still "Live" will draw an excessive current, while the "Dead" line will have normal or lower current flowing in the Motor's winding, and outward to whatever it can flow through.
Voltage will be near system voltage.
Proper fuse size selection will blow the fuse(s) in the "Live" lines.
Just my $0.02
Scott35