@HackedFridgeMagnet I haven't been able to find too much information on the communications protocol in use. It seems to be based on a TX and RX optocoupler at each end to manipulate the DC voltage of the line. The voltage is rectified from AC via a simple rectifier circuit on the outside board, with multiple diodes ensuring that the current flows towards towards the indoor unit.
Speaking of mains, the entire low voltage portion of the circuit (5v, 15v, gnd) is 115vac to earth.

not sure if that's a design decision or caused by something going wrong.
Thanks for the hint about checking the rail voltages. I ended up doing some further investigation and found that while they usually seem stable enough when the system is operating, it there are some circumstances where 5v and 15v sit at 0.7v.
Thanks for the offer with the scope, but, alas, Sydney is a little too far away

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@pepona A schematic for a similar circuit could be very helpful in determining how this one is supposed to operate. I'm also planning to look for some technical manuals for similar models of LG air conditioners - hopefully they decided to reuse the same design.
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After replacing a few capacitors on the indoor unit circuit board that seemed to be related to the communications circuitry, there still doesn't seem to be any improvement.
That being said, I think I'm getting a bit of a better understanding of how how the different parts of the circuit work and what issues it's experience (or at least the symptoms).
I ended up getting pictures of both sides of the circuit board and tracing out the connections between various parts of the circuit. Especially the power rails, as I'm trying to work out what generates and filters the power supply.
Screenshots:
https://www.dropbox.com/s/lbz1yn2gxbilhnb/board_bottom.png?dl=0https://www.dropbox.com/s/5f8m5o9906fvc9o/board_top.png?dl=0Red is 5v, Orange is 5v rail on the other side of a resistor (incomplete). dark blue is 15v. Grey is gnd (incomplete). Magenta is AC live. Light blue is AC neutral.
I also have a Google photos album of all the pictures I have so far showing more details:
https://photos.app.goo.gl/LUf87jHKtXdvBbU47-----------
One thing I did find was that the TX trace (visible under the led) had been corroded and didn't have any continuity - that definitely would not have helped here

. After scraping away the corrosion, adding some solder, and verifying continuity, it didn't seem to help at all

. Still a communication error.
I noticed this when checking the low voltage side of the TX optocoupler and found that one leg (anode) was connected to +5vdc via a 600ohm resistor, the other was supposed to be directly connected to the microcontroller (and connected to ground via a capacitor).
The microcontroller side (cathode) was measured at 6.3-7vdc to gnd, whereas the anode was measured at 5v. A reverse voltage across the optocoupler diode didn't seem right. My guess is that this voltage was measured due to the other end being disconnected, but I'm not sure if it could be some voltage from the transistor side of the optocoupler leaking through.
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There seem to be 3 different symptoms that may or may not be related.
1. Once the indoor unit powers on and provides power to the outdoor unit (verifed by checking voltage at the terminals), there is no sign of life or activity from the outdoor unit. There no led, fan or compressor, and both the 5v and 15v rails measure 0.7v at the test points. This seems to occur less often when I've been testing in the middle of the day so I'm thinking maybe it is related to temperature of the board or a component? . It may also randomly go back to this state for anywhere between a few seconds, to a few minutes, to giving up for the day.
2. (1) resolves itself after some period of time. There is then correct and stable voltage (relying on my multimeter here) on both the 5v and 15v power rails. The red led flickers rapidly, which I understand to mean that it is recieving communication from the indoor unit and is expected behaviour. However it will just stay like this for an arbitrary period of time. Sometimes only for only a few seconds before it starts up, but other times it could be tens of seconds, a minute, or never.
3. (1) and (2) both resolve themselves after some period of time. The fan starts up and then the compressor begins to come up to speed a few seconds later. Either the system will go back to state 1 after a few seconds (rarely), or the indoor unit will trigger a communications error (after exactly 2 minutes of turning on the indoor unit, regardless of when the outdoor unit starts) and the compressor will immediately shut off with the fan running for another ~30 seconds or so.
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My current next steps will be to replace some more capacitors on the outdoor unit which I identified as being part of the power circuitry (that circuit trace was very helpful).
One thing that I'm having trouble with though is working out how the microcontroller and low-voltage circuitry actually gets it's power. I see two possible candidates: the component marked POWER MODULE labelled "μPM1515-W" (have not been able to find a datasheet) below the 4 large caps, and the bridge rectifier and associated circuitry to the right of the 4 large caps.