It should trip the overload, but so should a phase loss from a blown fuse.
It is just another way of protecting expensive equipment. I can often protect many expensive motors with a single inexpensive phase loss monitor.
Should? Not really. Technically, it's not part of the job description for a basic overload relay. Some Solid State OL mfrs will ADD that feature, but if they didn't, they could still call their product an overload relay.
IEC bimetal overload relays CLAIM to "protect" motors from phase loss or imbalance, but not really. Here's how that all works:
A phase loss can be described as a full phase current imbalance, as unbalanced as it can get really. When there is any phase current imbalance, there is created, in the stator and rotor, what are called "negative sequence currents" that circulate in the circuits doing no productive work. In fact, the negative sequence currents create negative torque pulsations, so in essence your motor begins to "fight itself" a little bit. That then means that you have more current flowing into the motor adding heat to it, but not producing useful work. That heat is NO LONGER COMMENSURATE TO THE EXPECTED HEATING OF THE MOTOR DUE TO NORMAL CURRENT DRAW. It is superfluous heating.
Normally, an Over Load Relay senses current as a way to DETERMINE the heating going on inside of it, based on motor modeling design specifications. So as a basic rule, an OL relay does not know if the current flowing in is creating MORE heating than is should. Basic NEMA style OL relays are like this, they ONLY look at actual phase current flow. So under a phase loss or imbalance situation, if the current flow is still below the I
2t trip curve, it's happy and nothing happens. It has no idea that the current is only flowing in 2 phases and therefore is heating the motor up disproportionately to the standard motor model. That means a motor CAN BE THERMALLY DAMAGED under a phase current imbalance, and the OL relay may NEVER trip.
In the IEC Bimetal OL relays that supposedly have "phase loss protection", what they really have is phase current differential compensation. There is a spring loaded balance bar inside and the pressure exerted by all three current sensors is equalized against the spring pressure. If one phase is lost, the pressure is no longer equal, and the trip point of the relay is skewed to be lower than the setting. The point of this is to compensate for the added heating effects caused by the imbalance, but it does NOT actually "trip on phase loss" as they lead you to believe. The problem is, if the motor is lightly loaded and thus not drawing much current, this may not be enough compensation; the motor may still continue running on a phase loss and getting too hot even though these bimetal OLs are skewed to trip earlier.
As to Phase Monitor Relays, Phase Reversal is extremely uncommon unless it involves portable equipment or the use of emergency generators. So usually it's not worth worrying about. But one thing most of them never tell you is that voltage based Phase Monitors can be fooled by the regenerated voltage on the lost phase coming from a motor that was already spinning when the phase was lost. If you've ever heard of "rotary phase converters" it's the same phenomenon. Unfortunately losing a phase while something is running is the case more often than not. So a voltage based Phase Monitor only works reliably to prevent the NEXT restart, not necessarily to take a motor off-line WHEN a phase is lost. Current based Phase Loss / Imbalance protection is far far more reliable.
That's why I like Solid State OL relays better, and recent developments have provided options that make some of them very close to the same cost as bimetal anyway. In SSOLs, they look at the actual current in each phase and if any one phase drops below a threshold, they trip. Simple and reliable. If you use SSOLs you really don't need additional Phase Monitor Relays.