Wow... What a response! Thanks everyone. Fridges are obviously an area where the mad inventors get together and have a kitchen party.

It's not really a permanent install - I was just trying it out as an experiment (having already discovered that the start-up surge is right on the limit of what my 1kW inverters can deliver) and I'm not planning on disassembling my fridge-freezer.
Mine is a an electronic one (well at least the thermostat is a logic controlled one with a LED bar graph for the setting) so that probably rules out most of the ideas for hacking the thermostat / defrost timers, etc. It's a self-defrosting one and it does have a door heater as well, plus fans to active cool the radiator and circulate the cool air in-between the freezer and chiller compartments (it's an upright 50:50 fridge freezer). So maybe not your AAA rated fridge... although it is only three years old.

Here you can see my creation which I've dubbed "Fridgenstein"
. Sitting on the counter are some of my other solar powered kitchen appliances. I have a 400W toaster oven, a 900W kettle and (out of shot) a 400W rice cooker and (in a cupboard somewhere) a 200W automatic egg boiler (actually an egg steamer but it makes "boiled" eggs). The cooker, washing machine and fridge remain out of bounds for now
Although, I do run the cooker hood lights and fan from solar power.
One poster mentioned that if the inverter can't produce the required start-up surge, the compressor can fail to start and I did have this happen once (while doing the dishes in the evening). The charger was sitting at 15A and the fridge was doing nothing. The compressor had stalled on start-up, drawing 180W while not moving (hope nothing was damaged). I cycled the power and the fridge started up as normal after a couple of minutes.
It almost made it though the night on its dedicated battery (quite a feat for the 12 year old & often neglected battery). After sun-down, it ran on the house bank via the charger and then at about midnight I turned off the house inverter (with the bank depleted to 60% DoD).
At about 4am, I woke up and had a look at it... still working. The compressor was running. About 6am I woke again and this time the battery had died with the inverter locked out on a low voltage alarm. I fired up the house inverter and the charger sat pegged at 30A for about 30 minutes (with the compressor running) but I had to stop it as the house bank was now drained down to 80% DoD. The fridge carried on with its own battery and I let the sun start to charge the house bank.
It's not nearly so blue-sky sunny today so that will probably be it for the experiment. I'll let it run until noon and then I can say that I've powered a fridge for 24 hours on solar power but I also know that even a dedicated 2kW surge rated inverter isn't enough to guarantee that the fridge will operate normally.
That means my only real options are:
A SunnyBoy 1100LV grid tie inverter (these work at 21-60V so I can recycle my 600Wp 35V array).
A minimum 2kW house inverter (as this affords a 4kW surge and means I won't blow up the inverter if I'm running the 900W kettle at the instant that the fridge fires up). The house battery bank would have to be upgraded as well... 1kW is already a discharge rate of C/5 on my 220Ah 24V bank.
I'm not sure who to believe on the mod-sine debate. Some manufacturers say that it's ok to run fridges on mod-sine, some say not. Some manufacturers even say you shouldn't run fluorescent lights on their mod-sine products... Err... Hello... I don't have ANY tungsten lights in my house. All I know is that the mod-sine inverter I had made all my appliances buzz (even the lights) and almost caused a plug-in kWh meter to catch fire so I'll have nothing more to do with them :/
Then there's the solar input... I've been building up the system for about a year and I can now (on average) extract about 1-2kWh per day in the summer. I could utilise more power than that on long summer days but the fridge-freezer consumes a whole 1.5kWh per day by itself, leaving (on average) nothing for the rest of the house!
Looks like you need a hell of a lot of solar to run a standard fridge-freezer 24x7 (don't even think about the 365).
It's easy to rig up the spare kit I have for the odd really sunny day if I want to break a record on utilisation but, given our generally inclement weather here, I'll be sticking to the grid for my refrigeration loads for now.
The SunnyBoy looks like a better bet for my next big upgrade. It will plug straight into my existing solar array and it just plugs into an AC wall socket to feed the grid (no hardwire installation required). That can offset the fridge (and cooker and instant heating shower and immersion water heater) without having to actually deliver the power for those loads directly. Cookers and showers are immense loads but (like kettles) are very low duty cycle in a 24 hour period.
I don't know what arrangements there are with the UK utility companies for discounts / export tarifs. They have been very slow here to organise anything and if (as happened with Sharkey on the other forum I hang out on) the meter I have counts up regardless of whether the power is flowing in or out of the house, I could end up paying for the units I deliver to the grid! That won't do at all...