Complete Storyline #26: "Tic-Tac-Toe".
To unlock the storyline initially you will need to defeat the Sludge Mass. The boss of the Hazardous Sector.
Here is how I defeated the Sludge Mass:
*** Spoiler - click to reveal ***This is the equipment I used:
Hand: Assassin's Baselard
Body: Breezy Shirt
Trinket: Charm of Poison Resistance
The disease effect it gives you causes your max health to be capped at your current health which, combined with the poison it stacks on you as well, means your max health is constantly dropping whilst under it's effect. There isn't much you can do about the disease but having the Charm of Poison Resistance and an Antidote Potion significantly helps prevent your max health dropping too low too quickly. The trick then is to stack as many heals/regens as you can (eg, Health Potion, Cherry, Droplet of Healing spell) as soon as the disease drops off to get yourself back to full health.
Once you have defeated the Sludge Mass, to complete the storyline you will need to hand in 10 Cotton and 20k Water Essence to complete the storyline and unlock the Synchro mechanic.