Well a LOT is relative here. My tarp weighs 330g...
But today I was thinking do I even need a tarp at ALL?
Let's assume it's POORING down rain. I would clearly need some type of rain shelter.
But why not just have the entire hammock a shelter without an eternal tarp. Once you zip up you're completely in a rain shelter if the tarp were physically integrated into the hammock.
The one issue I can think of is if you have to leave your hammock in the rain. In that situation the rain would come into your hammock as soon as you unzip.
That's no fun.
So maybe more of a half tarp so that the entrance was protected from rain without the excess material.
... and I'm an ultralight backpacker. So yes I'm obsessing about weight here but that's what we do
Having a 12lb pack (with food) is nice I wanna be sub 10 though.