I need to find a way to allieviate compaction or find a groundcover that will tolerate it.

Some details:

- droughty California climate, zone 9

- yard is approx 500sqft, part sunny, part shady, and a little bit is deep shade

- has a large pepper tree in the center, whose roots are thirsty and EVERYWHERE, running under every inch of the ground, and start about one inch below the surface. They are so tenacious, they will even travel into drainage holes and up into pots!

- Tree roots leave the soil dusty dry no matter how much I water.

- the soil is so compacted you cant dig without breaking your shovel (done this a a few times now)

- terrible weed problem, heavy mulch does nothing (oxalis, bermuda grass, cheese weed, knotgrass, other weedy grasses, storks bill and more). If left unplanted it turns into a forest of weeds.

QUESTION: how do I break up soil compaction without harming tree roots? OR is there a ground cover that can keep the weeds down that will grow in terrible compacted soil?