Reducing weed pressure without poisons, tillage, or tarps

I was wondering what other folks here with large gardens and market gardens do to deal with unwanted weeds?

Do you have any good strategies for dealing with weeds both during the growing season, and outside the growing season?

This year we managed to keep some parts of the garden well weeded during the growing season, but now it is late autumn and very damp I’m wondering if the weeds are just going to come back to life if I rip them up and leave them on the beds like I do in summer.

I’ve also never quite got the hang of regularly using green manures. Does anyone successfully use these for fertility and weed control on a large garden without tarps or machines? What is your process?

In gardens with lots of annual weeds, which crops do you grow that will actually come up quickly and not get shaded out and taken over by weeds?

We have 3 large annual gardens. One of them is overrun with radish weed from me being lazy and letting weeds go to seed a few years ago (and I don’t like to eat radish weed). The second and third have sheep sorrel problems in parts, along with some other weeds around the place. Grasses can be a problem everywhere.

Similar problem here!  I've been letting grass grow to about `5 cm high then pulling it and leaving it in 2-handful bunches with the roots exposed. Left in rows while I do the next piece.  That seems to be doing the trick, none has come back to life.  Then I use the large leaves from the rhubarb as weed suppressors, aided and abetted by some shade cloth placed around the rhubarb plants - also aids with water retention during summer.  Elsewhere in the veggie plot, I use the hoe while the sprouts of things like cleavers and fumitory weeds are still small.  Any thistles I have to dig.
I should add that most of my nuisance weeds are the result of growing a trial "crop" to try out for natural dye - no bargains there, most made wishy-washy yellows!