If your client wants to build a trellis-like structure, I think it would work. I'm thinking it could be done very much like Native American sweat lodges are built. Those are usually made with willow twigs lashed together with twine, structurally similar to the concept of a geodesic dome. I just helped build one this past Sunday, and went through the whole 2-hour ceremony, sweat and all. Rained almost the whole time and the structure held with all the wet blankets, rocks and ponding water at the top.
The difference for your situation is that it needs a more permanent foundation, like concrete piers that support it at least 6" to 8" above grade. It's also got to be a whole lot taller than your typical sweat lodge.
If it's a large structure, I'd be tempted to build a simple framework of a more commonly used lumber (redwood or cedar), then fill-in with the eucalyptus.
Most of the eucalyptus species here in the US are disease and pest resistant, but I don't know about termites. If it were me I'd put in termite shields and poison the soil around the piers.
From my sister, the botany major, I have this info about common species in California. I haven't found any structural properties for these, however.
Eucalyptus fictifolia is also known as scarlet flowering eucalyptus.
E.citriodora has white flowers that bloom in fall and winter.
E.polyanthemos is also known as silver dollar eucalyptus and resembles quaking Aspen. This is the variety that you usually see as stands of windbreaks in wine country.
I'm actually starting to like this idea! Where's my sketchbook.......