Is poisoning that noxious weed sometimes the only solution?

The milky liquid and rash sound like it might be a rubber vine which is now classified as 'one of the most invasive plants in Queensland'.

Could this be it?

India Rubber Vine

Neko

That garlic mustard actually looks quite attractive in the photograph you posted. But what a pest! (I read the link.)
My rampant triffid vine is quite pretty, too. That's why I didn't remove it when it first appeared. And it was covering a bit of bare fence & trellis, too, so I thought: What's the harm? Little did I know it would be sprouting up all over the place before too long!!!

2,4-D.
That's the second time it's been mentioned.
I'm a totally ignorant when it comes to pesticides.
That variety is the least harmful to animals & the environment?

Thanks, J_B, but no, I don't think it is.
My flowers are much smaller & the leaves are more delicate looking. The actual stem of the vine is thin & rounded, but quite tough when established. Almost impossible to break or bend with my bare hands. They have to be cut. But the bit about the irritating sap certainly rings true! Honestly, in one of my triffid eradication missions I tried to cover myself from head to foot & still got a few bits of sap on exposed flesh. Horribly irritating! & the stains from the sap will not be removed by thorough washing of the affected clothes, either. Very stubborn, indeed.
It's interesting, I couldn't find any references to the (quite prolific) "fruit" (which apparently hold the offending seeds) in any of the links with plants that closely resemble mine. Yours is about the closest, so maybe it's a relative?

2,4-D

It's probably available at your nearest garden supplies store, Ms O, under a tradename for lawn weed killer. It's widely used on lawns, because it doesn't affect grass, and it's relatively safe, from what the site that the link takes you says. This is the stuff I recommend -- works better than Roundup on garlic mustard.

Thanks, neko.