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Use a dado the width of whatever thickness your boards are. Mark the center of each board, cut that far, carefully chisel the waste square on each, then slip them together. I made a wine rack like that a few years ago with 16 cubbys and it went easy doing it that way.
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A better alternative to making the joint "bang together tight" is to cut the joint as shown
here. It will prevent gaps from showing and allow you to assemble it without a great deal of force. Although the ends of the slots are shown square in the illustration, they wouldn't have to be. You could cut the slot with a router or drill a hole at the end and cut the slot open with a bandsaw.
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I think I would go with dowels.
The cross section would be three pieces - one long to fit the diagonal of the box and two shorter pieces
Line up and drill dowel holes
(the dowels pass through the middle of the long board and into the ends of both short pieces)
Make the entire middle section a little long
dry assemble the box and the cross shelf separately
lay the box on top of the cross section on a table
line up the corners of the box with the ends of the cross
then mark and cut the ends of the cross to fit the corners of the box.
Say a prayer
and glue it up.
Ray