I have this very annoying weed in only one flower bed (where I added mulch).
It has very deep roots so is almost impossible to dig up as even when I dig down it tends to break off.
What it does is send out several runners along the ground (in all directions) until a runner gets to a plant then it goes up the plant winding itself tightly around the plant.
It doesn’t appear to have seeds or flowers or pods.
It is not “dog strangling vine” I have that as well. The dog strangling vine is much easier to find and often able to pull out.
If you want I can send pictures.
Without the aid of pictures, I can only hazard a guess as to what your mystery weed is. That being said, I think you may have bindweed, Convolvulus arvensis. It can arrive in the garden within soil, compost or mulch, if there are viable seeds contained in these products.
I am attaching a couple of links below so that you can make a positive identification, along with information on how to control and/or eradicate this noxious weed.
Please note, in Ontario, there is regulation on chemicals which may be sprayed/used for cosmetic control of weeds. Therefore, we would recommend that you employ cutting, sheering, covering the offending plants or use of boiling water as a first line of attack.
If your invading weed is not Bindweed, please photograph it and send us in the picture. Good luck.