getting rid of nasty grass by organic means

Unfortunately my garden is infested with this pesky weed (please see the video https://drive.google.com/file/d/1Do_c4A1-4NWuU-B3MJdi1L7O13GvV8Xz/view?usp=drive_link). I think it could be called witch grass or couch grass -- not exactly sure but it is some kind of nasty grass. You can see the video of me attacking it with a hoe. I want to get rid of this weed or reduce its impact on the garden by natural means. I am trying to grow vegetables on that spot and the presence of this weed makes it a pain. Some people say that it can only be killed by spraying with a chemical but I do not want to do that. Any advice is most welcome.

Covering with something (cardboard, lots of paper, deep mulch) to exclude light for long periods of time is guaranteed to work. But it can take months to deplete the energy stores of the really persistent grasses like couch, so this means you can't do much with the space in the meantime. If you can build beds on top of what you're using to cover and kill the grass, then at least you still get to grow your crops.

That looks like vigorous and tenacious stuff. Unfortunately it's not possible to identify it from the video.

I think it would be very useful to dig a vertical survey pit in a patch of undisturbed grass, to see how deep the roots and rhizomes go. Know your enemy!

Perhaps it is possible to dig out deep chunks of sod and let them rot anaerobically, immersed in water. There is a lot of good biomass there. They can also be thrown in a half-barrel of boiling water for 15-20 minutes -- nothing of the grass will survive, but it will be raw material for building fertile soil.

A few hungry pigs may help in the fight, if you have access. Dig some dry corn in there and watch the magic.

I will second Douglas's comment on pigs.  Nearly anything organic that isn't toxic to pigs can be killed thru intensive grazing and adding some water.

Maybe use some tips from this thread:

https://permies.com/t/284233/permaculture/Success-battling-invasives-chemicals

I'm endorsing Phil's suggestion of paper/cardboard plus deep mulch.

But that is just the opening salvo in a long war. Your enemy wins a battle when you allow photosynthesis to replenish the energy reserves in the rhizomes. That energy is spent growing new rhizomes to spread out, and new leafy parts to harvest more light. You win a battle when you dig up rhizomes, and when you cut off the leafy parts as soon as they appear.

Deep mulch helps immensely because it forces the deep rhizomes to grow farther up before reaching the light, and it will colonize the deeper parts of the mulch instead of the hard ground. Rhizomes are very easy to pull out of the mulch, so you don't have to work so hard as you do in the video. You don't have to get the deepest roots out, but it shortens the war. With deep mulch the hard soil will be softer and easier to dig to get the last, deepest parts out.

A complementary tactic is to establish a barrier around cleared areas to block rhizomes from penetrating into the cleared area. Cardboard installed vertically in a slit trench can work, but will need replacing frequently. Sheet metal is a good choice for longer term, and very useful once you've cleared to the edge of your planned bed. Tilt the vertical barrier slightly inwards. The rhizomes will try to grow over the vertical barrier making it easy to pull them out before they make it over. Whereas the horizontal barrier alone allows the roots to encroach underneath.

Obviously, constant vigilance is the key factor. I have 2 year old beds that are completely clear, only requiring policing of the perimeter, and one year old beds that I'm still working on. With a garden fork, the roots are easy to pry out mostly whole, instead of chopping them into smaller pieces with a hoe or shovel. Smaller pieces just grow twice as fast as the original!

I have a 3 year old bed that I neglected, and the rhizomes are everywhere, but now clearing is pretty easy. The broken down deep mulch of leaves and wood chips has turned into fabulous soil that is full of earthworms and worm castings, and the rhizomes are easy to pull. I just pull back the top layer of chips, save the best soil underneath for the growing beds, fork out the rhizomes, lay down new cardboard, put the old wood chips back  to cover the cardboard and replenish with more chips and leaves to make it deep enough. With the summer heat, deeper is better, especially in the walkways. I can mound up the beds a bit, cover with 10-20 cm of good compost - its ready for transplanting. The walkways get 20-30 cm. This makes them safer to walk on as wet cardboard or paper with a thin layer of mulch on top can get very slippery, and hard rainfalls have a deep reservoir to soak into.

I wonder if timing may be a factor? For example, I notice that quackgrass rhizomes are easy to remove completely early in spring, before they have regrown the fine root hairs that died in winter. The root hairs act as barbs, and when you pull on them the rhizome breaks and makes new plants.

Thank you very much for the detailed replies. Part of the problem is that this garden is located miles away from where we normally live (8 hour drive by car). So I can't always be on top of the weeding chores. Also the garden was rototilled several times by a nearby farmer, and later on by myself. This definitely made the problem worse. The soil is quite sandy, which easily compacts and gets very hard in the dry season. The region has Mediterranean climate, which means no rain roughly between May through October. So the garden is irrigated with drip irrigation.

I really think that deep garden beds is going to be the answer. I am thinking of stacking three 2x4's along the length of the beds as border (about 25 meters = 75 feet). This is not going to be cheap for sure, but maybe I can think of a cheaper border alternative. Please let me know if you have better ideas to make the borders for the deep bed. Could borderless be an option? Then of course I need to add good quality soil and mulch. I also thought that if I had softer soil pulling this weed could be much easier.

I've got lots of couch, and its even more evil cousin bermuda grass. Plus now I seem to have kikuyu in a couple of beds, and to add to the misery mix there's no shortage of creeping buttercup and convolvulus :-( So I can definitely feel your pain.

I can attest to the long game approach that B Beeson talks about, too. As I keep piling on the wood chips, all the things I don't want become easier to pull out by hand because they're rooted more in the mulch and loosened soil that's forming underneath. I'm turning the tide for the most part. Sometimes I just cover a bed with some boards for a few months to get an advantage when I want to plant into it. Most of the beds in the original garden are 1x6 cypress and have decayed to the point where they're about to disappear, and in the newer cropping areas I'm not using any borders at all. Just piling on the wood chips.

I fight Bermuda grass the is a lot like what you are dealing with. Unfortunately I don't have good news. I like to grow in hugel beet style. I dig 2 to 3 feet down, this usually gets below all the weeds. I don't want to bore you with the details, but I add weed cloth to the edges. I make raised beds out of pallets. This keeps the Bermuda grass out for a while.
Even if I use weed cloth, or cardboard and 8 to 12 inches of wood chips it doesn't kill the Bermuda grass. I have pulled up weed cloth and found a thick mat of Bermuda grass roots. Sadly. You fight the fight, because you will never win the war. Just pull it as soon as you see it, and keep going.
Good luck to you.