Weed ID, please

Can someone identify this weed for me? I'm assuming it's a weed based purely on the fact that I didn't plant it and it's growing incredibly well.  Any self-planter that thrives is suspicious, in my mind.

It has a small leaf, with a reddish under-leaf colour. It forms a thick mat, up to ankle deep. Very shallow roots - pulls up in mats rather than clumps. It runs up/down rocks. At the moment it covers most of our second tier in the back yard - about 15 x 3 metres. It seems to prefer the slightly sunny sections; growing thicker where there's more sun.

Top side of the leaves

Underside, showing red tinge to leaves and stalks

Quite a small leaf

A section that I threw in the wheelbarrow

Growing up the rock wall. That's the toe of steel-cap in the corner.

The section Hamlin-the-Chook (as in I am the Pied Piper) and I cleared this morning. The weed is still thick to the left-hand side.

Edit: Someone suggested Wandering Jew (Tradescantia fluminensis). The descriptions I can find certainly fit. Seems it likes a bit of light; doesn't do so well in closed canopy forests - so hopefully smothering it with weed cloth and maybe some cardboard should kill it off completely. Not so sure how I'll tackle it in the retaining walls.