Answer: Eucalyptus trees don't make very good garden companions. The shade they cast, the litter they drop, and the fact that their surface roots will suck up as much moisture as they can find, makes successful growing of annuals and perennials directly beneath them difficult, if not impossible.
Wish I could offer some more positive advice, but planting anything under eucalyptus is chancy at best.
You may want to consider building raised beds and lining the bottoms with weed barrier cloth (to discourage root growth into the beds), then filling with topsoil and planting. This way you can control the moisture that your plants receive (they'll be first in line!) If you decide to build raised beds, most any shade-loving annual or perennial will be happy in the beds.
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