You want a system with a population of under 1 million, and no player factions. (I managed to do this in one system with a player faction, but they probably took their eye off it long enough for me to achieve my goal.)

It must have a native anarchy faction, of course.

Most systems look something like this on inara:

#1: 50% ish
#2: 20% ish
#3-#5 4-6% ish
Anarchists: 1%

(It always adds up to 100%)

Ideally, you want most of the settlements split between the #1 and #2 faction. #2 is your target. Make sure there are at least 8-10 settlements owned by #2.

Here's your first task: make sure you're running ED Discovery or a similar third-party tool, and take an Apex to every settlement in the system. You can divert after it's docked there, and before you disembark.

This will update all the settlements on Inara so you have a quick and easy record. (Updating Inara is also a great way to keep track of faction influence and conflicts)

Next order of business is to do INF reward missions for the #1 faction. Passenger missions, regular missions, ground missions, or turning in bounties at a station or port or settlement they control.

(Hand in 1m-2m of bounties at a time. Any more doesn't count as more influence so there's no point turning in 20m.)

Trading is meant to work but I've seen limited effect. EDIT: 5+ inf rewards were bugged at one stage, and gave no influence, but this appears to be fixed.

There's a limit to how much INF a faction can gain in a single day. Also, the value only updates once every 24 hours. I tend to do half a dozen missions in a 24-hour period, including courier missions which I deliver in the Apex.

As the #1 faction rises in influence, the rest compress down. What you want is #1 faction high, the #2 faction to hit about 8%-10%, and every other faction below 7%

#1: 70% ish
#2: 10% ish
#3-#5 2-4% ish
Anarchists: 1%

Now you can start working for the anarchy faction. Raise their influence rapidly, doing missions mostly (on foot or in space). You want them to reach the #2 faction so the two of them go to war.

#1: 70% ish
#2: 9% ish } WAR
Anarchists: 9% } WAR
#3-#5 2-3% ish

When the war breaks out you need to win one ground battle in all the contested settlements. This is why you want an out of the way system, because anyone fighting on the other side is going to wreck things.

Each day, after the tick, you can check Inara or in-game (station news board) for the progress of the war.

When the war ends, the anarchists will take over every settlement they won the most battles in. (Ship battles are irrelevant)

Now for the endgame - you have to slip the anarchy back down past the previous #2, which is now #3, without starting another war. (One of the two systems I was working in DID end up in a war. What I did was fight in space for the previous #2 faction, winning the war, without doing a single ground CZ. The anarchists lost that war but kept all their settlements.)

At this point you should have something like this:

#1: 65% ish
Anarchists: 15% ish
old #2: 8% ish
#3-#5 2-3% ish

DO NOT attack anarchist settlements here. You'd better hope that other commanders don't find the playground yet either, as it's all in the balance.

The easiest way to move the anarchists back down below 7%, without coming into conflict with the old #2, is to do all kinds of missions in the system targeting the old #2 faction, working for all the factions except the #2 or the anarchists. Strongly recommend you do missions where the givers are the #1 faction and the bottom two or three.

End up like this:

#1: 70% ish
Anarchists: 10% ish
old #2: 5% ish
#3-#5 4-5% ish

Now you can go play merry hell with the anarchist settlements, raiding them, looting them, taking their regulators and doing missions to bring down their INF.

I suggest you do missions FOR the #1 faction at the same time, to soak up the INF the anarchists are losing.

This is what you want:

#1: 70% ish
Anarchists: 6% ish
old #2: 5% ish
#3-#5 4-5% ish

Now you're safe, because the anarchists will continue to plummet towards 1% (this has already happened in Dainayiman), but there's no risk of them going to war or losing their settlements. Congratulations!

I've done this twice now, in two settlements simultaneously, and I have five more earmarked for future fun and games.

Recently I converted another set of settlements to anarchy, and I found a shortcut.

When the anarchists go to war with the #2 faction, win the ground battles in every settlement for the anarchists on the first day of the war.

Once the anarchists are 1 day up, start doing space battles for the other side instead.

The anarchists will lose the war but still keep the settlements they won in, which means they'll still be in third place after the war ends.