One caveat before I write this: I've fought all of 1 short multiplayer war in Civ VI, so I have no idea what I'm talking about. Please disregard this as the ravings of an armchair general:
One thing that was driving me nuts in the later turns of your war, suboptimal - that is, after your first army was wiped out - was that you made good use of the production bonus to churn out a huge army of horsemen - but you never massed them! It seemed like lots of times you'd just dangle them out there to get slaughtered by the Archduke, instead of pulling back to safety and trying to gather a larger force.
Here's some examples:
Here, Archduke's whole army is outside of your capital, apart from some scraps. Losing this city is basically game over (although honestly losing Mouse Spider was a heavy blow, too, and I'm not sure there's anything you could have done here), but you've got two archers and a horseman fortified up at Blue Ringed Octopus still. My assumption is that you were holding those garrison forces there in case the Macedonians struck for that city, but I think this was a mistake.
1)It's always better to concentrate your army instead of dispersing it. Defense should be done by units (or walls!) buying time until relief units can get there. It looks like Australia defended with two small garrison armies, each of which was overwhelmed separately by a Macedonian army not much larger than the two armies combined.
2)You can see where most of the Macedonians are, and know that if they move on BRO you can hit them from the south as they try to take the city. Thus, most of your units would be best placed at Irukandji.
Another issue I think I see is your horseman in this picture. What's he up to? He was fortified in both shots, which seems, er, suboptimal (couldn't resist). I think he got picked off by all the units Archduke could marshal on the other side of the river, and he didn't contribute much besides occupying space for a while. Could you have pulled back, given up the square right in front of the river, then hit whatever Macedonian unit crossed with all your might? It might also have failed, but I think you might have bloodied the Macedonians' nose a little more.
Moving on, the siege of Irukandji lasts 4 turns, while you had a nest of archers up at BRO. I know the archers wouldn't do much against most of Archduke's units, but they would have done somethin', surely. I woulda moved them up, instead of waiting for them to be next on the chopping block. Eventually, Irukandji falls, leaving you with this situation:
You write that you don't know which way Archduke is going to swing, so you're trying to cover both cities, but again I think this is a poor deployment to meet that threat. Again, you've got two smaller armies, most of them just fortified in place, neither one supporting the other. This means that the Macedonians could pick off each army on its own, and since they have better troops, with more experience, led by a GG, that's a very one-sided proposition. Instead, again I think you should ahve concentrated your horsemen (backed up by the archers) in the center, where you could then take them en masse to hit Archduke's army. Hell, if he went for BRO, you could even have lunged for Irukandji - not sure if that was a good idea, but striking back would have felt nice.
A couple of turns later, Archduke has made his intentions clear:
You write that you couldn't find a better place to stick one horseman, so you pushed him across the river. Meanwhile, Archduke's crossbow picked off another horseman you had south of the city. Again, I think both horsemen could have joined the modest army at Cassowary and formed a mass to hit Archduke's flank. As it is, watching this unfold it felt like you were losing horsemen as fast as you produced them, sending them out one by one to feed Archduke's army experience.
A few turns later and BRO's fall is near:
Notice that as the city falls, you had 4 units at Cassowary sitting tight, leaving the small army at BRO to be overwhelmed. I think that's the common theme in your defense after losing the battle at the frontiers - your units don't really work together, but instead fight their own separate battles.
Now, to be sure, you're facing an almost hopeless situation by this point. Archduke's army has more experience, he has TWO UUs on the field, and he's got a GG backing them up. But more importantly, he's using his entire army as a united force. Every unit pitches in, so he gangs up on one or two of your isolated units with his whole force, predictably wiping them out with ease, while doing the same thing on a larger scale with your cities - he isolates one and picks it off, while the defenders of other cities don't intervene.
Anyway, BRO predictably falls and this is the last tactical set-up of your field army:
I get that you have Alcazars here, but why is your crossbow on the front lines? Would it not be better to swap him with the horseman, so he can shoot out of the city while the horseman holds the fort? Anyway, you opt for a static defense here - you stick your units on the best defensive tiles you can, and just brace for the end, as far as I can tell from the reporting. At this point, of course, the war is lost regardless, but I wonder if fortifying in place was the best move. There're horsemen and other melee units in the rear that basically sit tight and wait to be overwhelmed - again, it's letting Archduke dictate the pace, and pick off units one by one. Thus, he conquers all of Australia with hardly any losses.
So yeah, to sum up: In your next multiplayer war (says the blowhard who's never fought a multiplayer war in his life), try to make sure your units work together. Keep them massed in an army, and don't disperse them in penny-packet defenders all over your borders. You had the production and the numbers to overwhelm the tough Macedonian army, or at least bleed it - but it looks like they never had to face everything at once. Instead, they massacred isolated units and took out cities one by one.
I think you reported a great game, and your economic development is impressive! I'm playing a SP game as Australia now and I didn't do nearly as well as you did developing it. I also think you have great builder/micro plans - your only real weakness, I think, is the tendency I noted to spread your units out and fortify them in place, instead of using them actively and as a united whole. Thanks for the detailed reports throughout and for being a good sport about everything.