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
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?