I pretty much agree with Red's accessment. I'm not nuts about a butt-joined "extruded" box meaning there's nothing preventing to chance of racking. That was one of the reasons I put a full inset panel in my center section and glued/nailed it to the gussets.
In your case, if you must have a full open back throughout, the L-shaped steel ¾" wide X 3" corner braces would work. You can also help things by gluing in a corner brace gusset, something like a 3/4 X 3/4 X 12" strip (or even quarter-round). You can recess it say 4" from the front for aesthetics, and if its 3/4 square, you can pre-drill it before gluing on 2-sides, then use say... 1" or 1¼" screws to lock the gusset to the top and sides.
While you don't have Norm's shop, you must have a drill, and from the bottom, you can drill into the side assuming the verticals all rest on the bottom horizontal plane. Use some thing like a 2" drywall screw (I used deck screws), and you can augment that joint with simple thru-holes for a flush dowel insertion from the bottom up into the verticals. Just be careful and use some sort of guide to make sure you drill straight. I know it's a lot to ask, but some sort of pipe clamp would do wonders for your cabinet's integrity when you're gluing it together, and of course... a steel square would be very, very helpful when you're assembling, gluing, nailing, and clamping.
Use a good grade of wood glue on the joints and keep a damp rag handy to wipe off the excess glue before it dries. You can use 2" finishing nails on the top and that should hold the box together pretty well. Always do a light pre-drill pilot hole (1/4" deep) so that you don't mess the wood up when nailing, and use a nail set to get that last 1/16" and the countersink.