The tendrils are effectively their influence lines. If your system is connected to their tendrils and they have pacific conversion, then it will eventually be converted.

If you want to stop them you will have to destroy their connecting system. Looks like that will be difficult in your case as it's being connected by two separate routes. The other alternative is to declare war, as this will stop any conversion from happening. 

In future, I suggest not letting them get inside your borders. Close the borders, and/or kill any vine ships that start connecting systems.

As a side note, it can be pretty fun playing as Unfallen, connecting all systems in enemy territory, then converting them all. I wiped out the Sophons in a game a while back in 1 turn.  :D :D :D

Hum...

I see... the issue is I already had the tendrils in since I created the colony... 

How the tendrils can be destroyed? If they can be?

Thanks.

I think the way to do it is to destroy the systems that are connected to them. One of them is Iaz, and the other one is obscured, but it's the one south of Rasam.

What's better, convertion works only while in peace or alliance... ba-dum-tss!

Yeah I think I'll have to boot them out.