You didn't specify which red wire was hot. Either way the circuit continues on beyond these boxes separate of the lighting control. I am going to assume that 'D' is the source of everything and comes from your house / switch.
Here is a series of tests, some of which you may have already completed.
In the box on the RIGHT
Power enters 'D' on red / exits 'C' on red.
>>> Test: with wire nut removed from the two red wires, the wire entering from 'D' alone is hot. (replace the wire nut when complete)
Power exiting 'C' provides power source for BBQ island or?
>>> Test:
can be tested later if necessary
Power from switch for lighting enters on 'D' black.
>>> Test: with the wire nut removed from the two black wires, the wire entering from 'D' black is hot when the switch is ON. If the switch is off, neither black wire is hot. (replace the wire nut when complete)
In the box on the LEFT
Power for the lighting circuit exits 'F' black and enters the box on the left as 'E' black.
>>> Test: With the wire nut removed from the black wires and the switch ON, only the wire entering 'E' is hot.
Power to the rear lighting exits ['A' or 'B'].
>>> TEST: Wire nut 'E' black to 'A' black then 'B' black one at a time and check which turns the rear lights on.
I will assume that 'A' black turns BOTH rear lights on.
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At this point I have to ask how you are turning them on. You said:
"The lamp posts in the backyard are switched on and off at a remote switch that is wirelessly connected to swimming pool equipment. It's located on the north side of the house."
Where is the switch (transmitter) and where is the receiver? Are you saying there is a remote switch (for example, near the house) which is talking to a receiver at the pool equipment or the other way around?
AND is this the switch you are using to power 'D' black?
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If 'A' black powers both rear lights then there is a good possibility that 'B' black powers the light atop the wall and maybe the front light. Alternatively I suppose it could power the transformer for the backyard low voltage lighting. Have you seen that anywhere?
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