Michael, welcome.
It can be difficult to adjust from UK-style gardening, I know.
Soil fertility is about nutrients, structure, fluids (both air and liquids), pH, antagonistic chemicals (eg. chromium), and the fungi and fauna that live in and on the soil. Depending on the size of plants, volume (effectively depth) of soil is also important.
One can analyse those, or one can be more practical and just see how things grow.
Wild plants grow well in the cooler months; witness the greening you mention. That suggests that the soil is already fertile - at that time of year.
You are probably wishing to grow plants just as well throughout the year. You could probably achieve that just by irrigating and choosing plants that would withstand the heat, sunlight and wind. But in discussion of mediterranean gardening there is often an undercurrent, as it were, of being water-wise. It could take a fair amount of water to irrigate as required, especially for a large area. And if plants did well, that demand could go up.
Perhaps you could consider what plants you would like to grow, and lean towards those that cope with mediterranean climates and ecosystems well. They often have lower fertility and water requirements. So your perceived need to fertilize could be less than you think.
...Just to suggest a way of looking at your problem.
Best of luck and enjoyment.