Mystery plant, help please

jjwkirkburtonKEpwWaKl

I have this stuff growing across an entire large flowerbed. Very distinctive fleshy tubular roots,  growing everywhere about 4-5" down and producing thin white stems with green shoots.  Concerned I must just spread it further if I try to dig it all up. Any ideas what it is please? 

Busy-Lizzie

It looks like the weed I had and asked about in this thread https://forum.gardenersworld.com/discussion/1051368/does-anyone-know-what-this-weed-is-please/p1

Turned out to be perennial sowthistle. I have dug loads out but it still keeps coming.

Posy

Could it be alstroemeria?

jjwkirkburtonKEpwWaKl

Thanks both, appreciated.  I will check those out.