The minimum wire diameter is determined by how much voltage drop your application can tolerate. Calculate the maximum wire resistance you can tolerate then look through a cable manufacturer's catalog for a cable with a low enough resistance per meter to be under that after multiplying by the total wire length there and back.
While straightforward to many of us, I was concerned that this was too much of an ask from our OP. So, rather than set them on a difficult task, I had a look at the cable they have at hand.
From the photo, it would seem the OP is using a mains rated cable - thick white sheath with red and black insulated wires within - but that it would be at the lower end of current carrying capacity. My guess is it would be for lighting circuits. Also, at 5 ohm for 140m it works out as 0.036 ohms per metre. With 10A current that is about 0.7V drop per metre of twin cable. So long as the cabling throughout the supply is of
at least this gauge and the total length from the transformers to the load is less than, say, 2m I can't see a 1.5V drop being a problem. Going heavier gauge is never a bad idea - but this should work for testing.
Questions to OP: 1. Is all the wiring connecting all your components of this gauge or thicker?
2. What is the total length of wiring from the transformer to the load? (Just answer for a single wire - and we will double it for any maths.)
How is the circuit constructed, under the cartons? Is possible to show a photo?
This is something I have wondered as well. In the photo, I don't see any soldered joints, just the terminal blocks used to connect the transformers. How are the connections we can't see being made? If you are simply twisting wires together, then there is going to be a lot of potentially poor connections, especially for pulling currents of 5A and more. Heavy current connections need to have very good contact or you will lose a lot of power. The terminal blocks are a good move, but what else are we not seeing...?
Request to OP: 3. Can you please show us a photograph of the hidden wiring?
It is not clear where the measurements are done. Are these on the controller or at the output of the circuit?
This has been a problem from the very outset.
Where you measure a voltage is important and for higher current situations can be absolutely critical for understanding what is actually going on. There is also the issue that we are not seeing the complete setup, so there may be some hints or obvious errors that are invisible to us.
Request to OP: 4. Can you please provide a schematic of the
entire circuit - from the supply, through the controller to the load? (Do not try to look inside the controller - just show the connections to it.)
One more test that you can do, is to try to measure the ripple voltage of the power supply. Set your meter to AC voltage, measure at the output without and with a little load, on each of the four and the entire combination of them. [with capacitors connected]
This would be an interesting - and useful - test.
With no load, we would expect this AC voltage to be pretty small and, essentially, insignificant. When this DOES become interesting is when you put a load on it. I would be interested to see what it was like at around 1A load current. The actual current isn't hugely important - but if you can measure it accurately, that will help in doing some math. Oh, and don't forget to do the DC voltage measurement as well.
Request to OP: 5. Can you test the supply as suggested showing:
a) DC volts, AC volts - with no load and..
b) DC volts, AC volts, DC amps - for a load current between 0.5a and, say, 3A