Could be couch grass which is as you describe. Or it could be seeds blown from a local farm crop. More likely the first though and it is as tough as old boots. Constant spraying is the easiest option but you still have to be vigilant.

Sorry A, surely that's not couch grass? though I don't recognise it

Looks like bindweed, Convolvulus arvensis. Does it start to twine and climb?

Doesn't look like bindweed to me ... but it does remind me of alstromeria. Have you ever left some to see what it does, in terms of growth habit and any type of flower?

Whatever it is, the roots have obviously spread far and wide, and you're not getting right underneath them to get them out - needs a full sized garden fork or spade. If its mixed in with other plants, then if possible, wait till autumn and dig up perennial plants, extracting these roots as you go, put them to one side out of the sun, dig the whole area, deeply to get the roots out, then replant everything. You can't do this with large or well established shrubs though...

it reminds me of the orange alstomeria that I have. it has white fleshy parts too and the leaves look very similar.
they respond well to Glyphosate weedkiller.

Firstly, you have not dug them up, Graeme, simply broken them off. I also go for the orange Alstomeria which can be a pain (unless you like it). It grows from small white tubers very deep in the soil. Glyphosate, Sbg, must give that another go this year!

Just to say many thanks for all your kind comments and help on my problem......concensus seems that it could be
Alstromeria.
I will pot one up and await results with anticipation.
I will keep you all informed.
Many thanks.?
Graeme

I

Ihave just spent an hour digging up roots.....rhizomes from the aggressive plants.....weeds growing in my garden.
I think that this is only a fraction of what is there.
These have come from approx two sq metres, around eighteen inches deep....I'm nackered ....worried that I can't get rid of them...TRIFIDS.
HELP!!!!.....see photo.
Graeme.?????

that is definitely alstromeria. if you don't want to keep digging resort to weedkiller they will respond well to it. go for a systemic one so it will get down and kill those white tubers.

You need to let the leaves grow - glyphosate works 'through the green' so you apply it to thriving green leaves and it works back to the roots. Eventually....