Your question mentions that you only have a circular saw. This makes me question what other tools you might have access to. Be aware, though, that you can make jigs to help you with some tasks.
This suggestion would require the following tools:
- A hammer
- A bench vise
- A drill
- A looooong drill bit
- A hacksaw or angle grinder with cutting wheel
I would suggest you make some internal reinforcement. Several people have suggested long screws or threaded rods. Have you considered installing a "hidden" support?
Imagine you purchase some 1/4" steel round bar. Now cut it to length and bend a right angle into it, making an ell shape of rounded rod. Make two of these per shelf, times however-many shelves.
Now, using your circ saw (and a narrow chisel, if you have one), make two grooves in the back (wall-facing side) of the spine. The grooves should be 1/2-3/4 inch apart, centered left-to-right. The grooves should be the width of the rod, and the depth of the rod. They should run from the bottom of the spine to the height of the top shelf.
At each shelf, drill a hole inside each groove. The hole should be just slightly bigger than the rod. The hole should extend through the spine, into the shelves as far as you can possible make it reach. (For 12 inch deep shelves, see if you can make a 10" or more deep hole- they sell long drill bits.) The holes should be parallel to the shelves (obviously! otherwise the holes will come out the top/bottom) and at an appropriate angle to point at the front corners of the shelf. I believe this is about 55 degrees from the centerline of the shelf.
The idea is to insert the bent ell shapes into the groove in the spine, and into the hole in the shelf. You can then secure the bottom end of the ell shapes using epoxy or just screws and metal tabs or just screws carefully placed. The ell shapes are totally invisible from every direction except when the shelf is pulled away from the wall. They're customizable to the specifics of your shelf, and they should be something you can make with a few tools.
My local orange box sells 1/4" x 72 round rod for $5.87 (USD). Your drawing suggest a small distance between shelves - maybe 6"? So 18" of rod would bend into a 12" leg and a 6" leg nicely. Thus, you could get 4 ells, or 2 shelves, per unit. 5 shelves would cost < $20 to reinforce, plus any screws, drill bits, and other tools.