Growing near gum trees

If you can get hold of some lilly pillies they might be worth a try - they seem to be pretty hardy if you can get them going

Eucalyptus are allelopathic ( en.wikipedia.org/wiki/Allelopathy ) which is nerd speak for "a plant that wages chemical warfare on other plants". Unfortunately it isn't just Eucalypts either, Australia is full of these trees. That is kinda our thing... Africa has nasty parasites and Australia has nasty trees.

With thorough composting you can use Eucalypt leaves and bark on the garden, but otherwise clear all litter or it will harm growth. Eucalyptus roots do the same but that's hard to avoid (raise maybe?). Fresh leaves and bark are the worst.

If Eucalyptus trees could talk then they would spend all day yelling "F_off" to all surrounding plants. And they do, but with volatile chemicals instead of sound. That is one of the reasons if you look up into a mature canopy, its like each tree avoids the next (it's not just about light).

Edit: as mentioned by others above, native Australian bushtucker plants would be more adapted to deal with it.

If you are going to use any sort of raised garden beds in the area of gums as per the photo make sure they fully enclosed or else you will end up with a bed of tree roots,

When building a fully enclosed bed have an outlet at both ends several centimetres above ground level so that excess water, like after a heavy rainfall, can escape and not water log the beds.

The sunlight/shade as depicted in the photo, I would say, would make a good area in the summer for growing leafy plants as they would tend not to get burnt as those do that are in direct sunlight.