Frostpunk

Make The Other Three Settlements (Children's Mine, Shipwreck Camp, Hot Springs) Distrustful.
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I Did This At The End Of My "Survive 100 Days On Extreme" Playthrough.
Look To A Different Guide For Help Surviving Up Until This Point, But Needless To Say -
If You Have Survived Well For 100 Days On Extreme, Then You Should Also Be Self Sufficent Enough To Start Producing Extra Supplies For The Other Settlements.

Do You Have To Play On Extreme? God No! Just Play On Endless If You Feel Like it.
-With Your Endless Game Wrapping Up, You Can Look To Some Achievement Clean Up
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Endless Slave Driver
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Set-up
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Start Sending Out Scouts (If You Have Not Already) With Upgraded Sleds -
Your Goal Here Is To Find More Steam Cores, To Upgrade The All Settlements To Level 2.

Start Checking Off The Stuff They Need -
Example:
"Helping Hands II - Send Us 10 Workers, 30 Wood And 50 Steel"

- Doing These Things Does Not Increase The Amount They Like You.
- Neither Does Sending Them Emergency Aid, Go Ahead.

After You Do A Few Of These, If They Have Left Over Favor Spend It. (Get More Workers, Coal, Food; Whatever You Need)
-This Also Does Not Affect Relationships.


Getting To Distrustful
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About A Day Before Each Storm, And (Give Or Take) A Day After - The Settlements Will Talk To You.

There Are Two Types Of Events Here -
Settlement Disputes
- One Settlement Will Ask For Help Against Another.

Example:
Children's Mine Has Found A Steam Core, But Hot Springs Says It Is Theirs.
You Are Given Three Options Here-
Give Steam Core To Children's Mine.
-Hotsprings Will Be Displeased.
-Children's Mine Will Be Pleased.
Or
Give Steam Core To Hotsprings
-Children's Mine Will Be Displeased
-Hotspring's Will Be Pleased.
Or
Actually It's Mine.
-You Get Steam Core.
- Neither Town Likes You More, But Besides Exhausting Favor, Does Not Change Relationships With Either Town.

The Correct Choice Is Always Take The Thing They Are Fighting Over, For Yourself.

Settlement Asks For Help
- One Settlement Will Ask For Your Help, But It Only Affects Them.

Example:
Shipwreck Camp Needs You To Help Them Get Rid Of Evidence Of Their Revolt.
Two Options Here -
One Of Them Will Displease Shipwreck Camp, While The Other They Like.

Pick The Option That They Do Not Like. (The One That Displeases Them)
Picking This Option 3 Times Per Settlement Will Get Them To Distrustful.


Advice On Event Spawning
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Towards The End, I Had 2 Settlements At Distrustful And One Settlement At Neutral.
It Kept Giving Me The "Choose One Or The Other" Event Pre- And Post-Storm.

If Its Been A Few Storms And They Keep Giving You The First Event (Bad Event) Rather Than The Ones That Let You Lower Their Relationship Level -
Upgrade The Settlement That You Want To Hate You To The Next Level.

Example: Raise Children's Mine Level From 2 To 3.

When I Was At Children's Mine Level 2, Hotsprings Level 2, And Shipwreck Camp Level 4 -
Enough Events Had Spawned To Get To Distrustful.

I Finished Around Day 176, The Achievement Popped A Couple Days Later.

Hope This Helped!
Good Luck!

Not a guide but advice on getting it.
Load up that old save game and you should get it.

I have tried for about a month, It seems it was silent-patched sometimes around the 24'th Oct or something similar because now It unlocks roughly 2-3 days after the last settlement is made Distrustful.
Try to constantly call the three settlements and ask for stuff, maybe that helps. For me It unlocked on the screen chat with the Lumber Camp when they were telling me they don't need anything more.


/Credit goes to GUNGA on the steam forums for suggesting it.
Good luck,