I cut the back panel of both frames in half lengthwise, then cut the resulting 4 parts to size, so they could serve as “tabs” to hold both frames together. I painted the cut edges black.
I glued 2 “tabs” to the inside of each frame; one horizontally, the other vertically. When both frames are joined, friction holds the resulting box together.
Before gluing the vertical tabs, I drilled a hole on each side to hold a bamboo skewer (also painted black) The skewer is flexible enough that it can be fitted in the holes without glue, making it possible to disassemble the display.
Test-fitting the skewer…
I squeezed the keychain rings until they fit snugly around the skewer, keeping the keychains in place even when the box needs to be moved.