How can I secure this bookcase to the wall behind with me opting not to use a backing piece as to show the coloured wall behind?
Generally for this sort of thing you have quite a few methods to pick from, but a few of the usual options1 are not really viable if the bookcase is resting on top of the cabinets, and I think from your description you don't have the space either.
But the bookcase being supported in this way is very beneficial since it takes almost all the strain off any wall fixings, almost to the point where you're only attaching it to the wall to prevent the possibility of it tipping forward.
So three screws into plasterboard/drywall with suitable wall plugs, or two screws going into studs or a solid wall will be perfectly adequate here. But obviously use as many screws as you need to feel comfortable.
The first option I thought of is a single ledger board/hanging board, which you'd simply screw to the wall. This doesn't have to be mounted under the top as is usually the case, it could usefully be placed under the second shelf down (three up from the bottom) where it could also act as a shelf support — you could make this one of the 'heavy shelves'2.
Another option are the very old-school brass hanging plates:
Since I think you have no space available at top you'd install these at the back of either of the two upper shelves. It should be easy to arrange shelf contents so these are hidden from view.
Obviously any similar hardware with two or more screw holes could be used instead. I didn't initially think of them but this could include figure-8 fasteners which are very small and unobtrusive, but due to their thickness you may want to route/drill/chisel out a shallow recess to mount them into.
1 Including keyhole hangers and what we now call French cleats.
2 You could add this under every shelf so they match visually, which will simultaneously remove almost all worries about shelf sag.
