I think this is a Phasmatodea (also Phasmida or Phasmatoptera) but I'm not sure because I couldn't spend time with it. Its tan body was 7-8" long and about as hefty as a pencil, with small nodules and bumps that made it look amazingly like a stick. Its antennae are about 4" long. In the photo with my hand, my fingertips are about half an inch from its head to show size. It hung motionless from the rafter for about an hour, with its two central legs raised upward, above body level, just like tree twigs would be. Very convincing.
The habitat around where this walking stick appeared is deep, relatively undisturbed old rainforest, in the foothills of the Maya Mountains.
This insect was very calm, and although we didn't handle it, it didn't seem to mind/notice fingers nearby. A smaller one was spotted the following week in the same vicinity.