Gorgeous Mediterranean area, and I can imagine just how stunning it would be with a fire pit in winter time. To keep with the style, keep sun out yet avoid creating an enclosed feel shade cloth is the best way to go. If you wanted something easy to put up and take down, I agree a gazebo would work really well, like the one below: These are some DIY solutions - you can usually pick up these products from your local Bunnings, or learn more about them online. Plenty of colours, sizes and shapes to suit your space, too! Hope helps - can't wait to see what you decide to go with!

Thanks it is the biggest covered patio i have tackled + we had to follow the angle of the existing pool. Nearly had a mutiny from the roofies when they saw the length of the roof sheets.

what an idyllic place to live (despite the cyclones) and anything that keeps the hot sun off the concrete floors will help keep the area cooler so extending the roof over the whole area would be an excellent idea and consider fixed shade cloth screens on the western side to block hot sun and provide some protection from wind and rain without obscuring the view

I know you said that you don’t want to paint the brick but between the ceiling, brick and paving there is just so much brown hard surface which will work against your goal of a cosy feeling. Do you need to keep it fully roofed? I’d suggest some outdoor rugs, in a lighter colour, to define the seating and eating areas. Timber or wicker furniture def no glass or metal. And a lot of plants and greenery. To get comments of layout you need to post a sketch of the floor plan of the space with measurements.