Joining plywood? What kind of glue?

I'm going to be using 1/2 plywood for my build and have worked with it plenty of times, but never in a case where I want to join pieces perpendicular to one another without using an interior brace on the join. Usually I would put a block of wood on the inside of the join and and screw into both sides.

In this case we want to join it without any interior wood holding the pieced together. After some research it looks like the way to do this will be using rabbeting around the outside edges and gluing the pieces together.

But what kind of glue to use? I would think something much thicker than typical wood glue.

Is using rabbets and glue the way to go, and if so, what kind of glue have people had success with on these types of projects?

I'm thinking about using some metal brackets with screws around the outside edge to hold things together while the glue dries. I'd probably leave them on long-term as well.

My builds I've just predeilled the plywood and used wood screws. Might've used wood glue on the first one - I can't remember. No rabbets, just butt the pieces together. Helps to have extra hands to line things up