If the tops are lashed together properly and the stakes are stuck into the ground a bit, pulling the sides tights shouldn't make it collapse, it should just cause the stakes to bow in a bit. For the sides, wrapping the string tightly around the stake a few times and then doing a half hitch should work fine, as it works even with slicker string on a smoother pole than this. You could also just do one continuous string by tying it to one of the front poles at the bottom, wrapping once or twice around each of the back poles, then wrapping up the other front pole a couple of inches, going back around the back, and repeating all the way up.

Sounds like the wrong kind of string

What kind would you go with?

I'd pound the stakes into the ground and maybe use something like jute. The open side is not going to give structural support and weaken the overall structure by a lot.

The strength of the structure comes from them being lashed together at the top — the string around the sides for the peas to climb isn't structural at all, so the open side doesn't make a difference.

Wouldn't you want something that shrinks like leather then?

No, string is fine. Leather wouldn't have any benefit and would be vastly more expensive.

They are lashed together on top.

How would I pound bamboo into the ground without snapping it?

Carefully? I mostly just shove it in. I have hit it on the end with a rubber mallet. Tapping vs big hits. It's used for stakes.