I have such an air to water heat-pump. It reaches 55°C without the supplementary electric water heater, which is more than sufficient if you have a hot water tank. The supplementary electric heater is really only used when too many people want to shower in a short timeframe or to occasionally heat the tank up to 70°C+ to kill off any bacteria that might be growing in it.

I don't think you will find a heat-pump with a variable drive compressor like you are proposing, but a DIY solution might be feasible.

But why not just install a vacuum tube solar-thermal water heater? They are more efficient anyways.

But exceptionally, I’ve heard that a PV can directly power a compressor with no middleware.

I don' t think I've ever heard or read that about compressors. I also know that brownouts aren't good for them since they have to work harder by pulling more current to compensate for the reduced voltage. That causes them to run hotter. It's also not taking into account any controller electronics that may be in the mix which are also sensitive. Granted, this is all based on knowledge from refrigerators, but the components are basically the same.

However, I have read people running the electric heating elements in hot water heaters directly from PV, either directly or as a dump load. If I recall, the thing to watch out for with that is the thermostat; they're not made for DC and can fail to work properly. Not sure what the fix is for that (possibly replace it with one rated for a larger AC current, but I don't recall).

The electric elements are just "dumb" resistance loads and generally don't care what kind of power goes into them. Compressors are quite a bit more complex, so I'd be wary about feeding them anything but the "type" of power they expect.

Why not thermal solar panels wired to a water heater? Seems to be the least energy transfers, each of which incurs efficiency loses

instead of powering the heat pump from the wall, the heat pump can be connected directly to a PV

I have no experience with this exact combination. I know that "batteryless" inverters exist, but most of them are on-grid inverters. In that scenario, all that matters is monitoring your production: if you don't want grid energy, you only run your system when your PV produces enough.

Another type of batteryless inverters are "pump inverters". Farmers seem to like them for pumping water from wells into water towers. A pump inverter can be configured to run at 50 Hz (or 60 Hz for North Americans) and 230..240 V (or 110 V for North Americans) alright, but it is not designed to power electronic devices, but dumb agricultural motors. There is considerable risk involved with powering a heat pump from a pump inverter, unless you find an exceptionally simple and dumb heat pump with very limited or resilient steering electronics.

Efficiency losses are small anyway, but mostly happen during battery storage or when voltage needs to rise or drop considerably (e.g. a transition of 700 -> 24 V or 24 -> 240 V would cause a small efficiency loss).

I’ve heard that a PV can directly power a compressor

This seems unlikely as the compressor would have to be a brushed DC motor. That kind of motors don't last long, they wear out their brushes. Long-lasting motors are brushless, and those generally cannot be run on DC power. For example, a "brushless DC" motor is essentially a three-phased AC motor, just its controller (full of smartness and MOSFET transistors) accepts DC input.

If you have a good technical overview of your heat pump system, maybe you can locate a point where regulated DC can be fed into the system, but that would be hacking. Alternatively, maybe a niche market already exists for DC-powered heat pumps, e.g. for caravans, trucks or ships? But on niche markets, prices typically aren't good for you. :(

I think the cost of all the additional wiring and components would outweight the benefits.

What if the sun isn't shining? You would need a heat pump that can use AC and DC. What if the sun is shining, and the water is hot enough? Your PV would need DC cabling and a converter to AC to send power to the grid. And all these components would need some switches etc.

It's just easier to have all your home run on AC.

IIUC, it’s variable D/C which must be regulated and/or inverted to A/C involving more hardware, conversion, and waste.

Actually, I had a discussion about this with someone who really knows this stuff recently, and we figured that the inversion process isn't actually that inefficient. The efficiencies achieved are often >95%, so there isn't actually a lot of loss.

There are 'hybrid' heat pump mini splits (for air conditioning / cooling) that take both AC and DC input. (See airspool, EG4, and hotspotenergy)

All of these are configured to have dc compressors and AC to DC inverters.

If you have a hybrid system, you either take the conversion losses on the AC or DC input.
For aircon, having the losses on AC to DC conversion makes the most sense - if it's hot out, it should be bright out and there should be plenty of DC solar output - and topping off with a little AC and taking a loss should be fine.

I don't think this is the case with water heaters on average - if it's cold and cloudy or night you're still going to want hot water. I think it makes sense if you're in the desert - but most people are not going to be. So don't think this will be done commercially