Actually, this is a huge difference between deamons and dark eldar, both live in another dimension, but main differences: dark eldar can come ou when they want, they don't need a psyker or any help and they're fine in this reality. Deamons can't stay in this reality unless they open up a permanent rift over a world that infests this reality to form a realm that's kind of in between both.
In short dark eldar can stay in this reality without being affected like you think. When you're in the webway or in this reality, you still need a dose of souls.
Now Commorragh doesn't represent the whole webway far from it, and it's not the only city either.
You do have eldar type of cities, one is mentioned in the Path of the Outcast. You can also simply have a base somewhere in the webway or you can just hang around in the webway, it's kind of like a network of caves.
Psykers attract deamons, other races who are connected to the warp attract deamons, it's no coincidence that psykers aren't allowed in Commorragh.
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I was just under the impression from Path Of The Renegade/Incubi, that any small amount of time spent outside of the webway was very draining upon the Dark Eldar.
It depends how much time and what they do in this reality. If these corsairs are very active, of course they're be drained. Fighting makes you tired, but instead of just needing food like normal races, dark eldar also need souls and suffering, once they killed a lot, they come back rejuvenated to COmmrragh, those who didn't kill much come back drained it's to be expected.
In Path of the Outcast you have corsairs made out of a mix of both eldar and dark eldar, the leader is an archon though, but the draining is never mentioned, dark eldar can remain in this reality as much as they want. Still full dark eldar corsairs should still spend quite some time in the webway, even if it's purely for travelling faster or just staying somewhere safe.
Outcasts like The Duke stay out of Commorragh, but they can still spend most of their time in the webway if they want/need.