Hmm, they are robust..but I figured the mulch would do them in, I can't imagine hardly any seedling germinating and growing through a solid foot of mulch. Course, if you're getting fresh seeds on top of it...not much to be done about that.

Since they're annuals, best to eliminate everything as soon as possible, and rip every vine with flowers out before they reseed.

Is it morning glory or is it bindweed?

looked up bindweed and that sounds like it. Except it has white flowers not purple.

I think of bindweed with white flowers...so you're probably onto something...that's NOT an annual problem and yes, it'll come through mulch.
Herbicides...I'm afraid.

We've seen it grow through fabric and xeroscaping rock.

Bindweed is considered a noxious and pernicious weed in most states. It is actually mentioned in the state law. A city will have its own law, but the state law elaborates on any city law usually. You can get a fine for it, REALLY. In most towns you can take a picture of it and send it down to the housing code inspectors and they will send someone out who will issue a warning to the homeowners . Now if you want to kill it , you must remove all the soil and put it through a sieve to eliminate any of the white root of the weed. It is very bright white and brittle. Any piece will grow back a full plant from the little root. It will be even more vigorous. Expect it to come back and dig it out completely . I take out a margin of a foot around it like it is cancer. Which it is. Some homeowners think its pretty and contaminate the whole neighborhood with it.