Polypro or nylon mason twine is really cheap and will last a long time. Just go to a big box hardware store and they'll have loads of it, in many colors. Actually, for outdoor use, nylon is more UV resistant than polypro. "Tooth for the veggies to grab"? Veggies with tendrils can grab anything. Teeth not required.

I use string trellises very similar to yours. As strong as synthetic twine is, some plants (such as various beans) are reluctant to cling to it... yardlong beans in particular. I use that synthetic twine for the horizontal runs between poles. I too did not like jute, it seemed to deteriorate too rapidly. Sisal baling twine has worked very well for the vertical strings; it is rough enough for plants to grip it tightly, and everything I grow accepts it.

Baling twine comes in several grades. Only the lightest grade has broken during the season, and that was during a storm. I use the next grade up from that, and it supports anything that climbs, including cucumbers, tomatoes, and squash.

Baling twine comes in large rolls, and is very economical - but perhaps impractical for a small garden. You would, of course only need to buy it once... and there's something to be said for that. ;-) For smaller quantities, you might want to use a heavier cotton twine; but in a wet year, that too might have some failures.

Edit: since your trellis is a fixed length (based upon the raised beds), that opens up several other possibilities. One might be to use steel concrete reinforcing mesh (remesh) instead of string. It is similar to welded wire fencing, but ungalvanized, and with wider gaps to allow easy access when picking. Because the steel is ungalvanized, it does develop a coating of rust... but that provides a rough surface to cling to. Remesh is strong & lasts virtually forever, I have several remesh trellises which have been up for 10+ years.

Yeah, actually I have poultry netting on all of my trellises, and it works great. No weaving necessary. You can get galvanized stuff. Lasts forever. Unless you want to hang squashes and melons from it, it'll work fine.

I use nylon netting designed specifically for climbing plants or nylon twine. Lasts forever. Well......at least the 10 or so years that I have been using it :-)

I dislike plastic string in the garden for both aesthetic and environmental reasons and always use jute or sisal for tying in plants. I prefer it as it is biodegradable. At the end of the season it just goes on the compost heap along with the plants it supported. However, since your frame appears to be permanent I think zeedman's suggestions would be best in your situation.

I have always used netting from Gardeners Supply--I think it's jute. Works great and there's enough for several years worth of cuke trellising--you just cut it to the size you want. It's fully biodegradable.

I think biodegrability is indeed an asset, but for things that you want to last, it isn't really. Biodegradable shovels? Biodgradable hoses?

Of course, Dan. That would be ridiculous. But for things like string, plant ties and other sundries I prefer not to use plastic. It just adds to the unrottable junk we leave around the earth and I dislike the appearance. Did you read my final sentence? I said I thought that metal mesh would be best for the OP since s/he has a permanent structure. Personally, for beans I use bamboo or hazel rods tied with jute. For peas I use pea sticks. For cucumbers bamboo tripods. I can't grow melons in my climate.

I mostly use deer netting, have had it up for several years now. Use it for beans, gourds, peas, etc. I have also used that orange snow fence in the past as well as chicken wire.

Wow, thank you all! Sorry it took me so long to get back.

I think I'll try some sisal baling twine next time ( Hat tip zeedman).

Thanks!

You can buy hemp twine up to 170lb break strength, it's pretty cheap too. basically the same as sisal but a different plant. The fibers are rounder so the twine is rounder, possibly more consistent (less thick/uneven bits) than sisal; which could work in the advantage of sisal.

I don't use twine for your purposes (at least not as large) so I will not recommend any specific products.