[Arran]

Is the veneer delaminating from the solid wood at all? This sort of construction was very common for constructing solid wood parts on radio, and other, cabinets in the days before plywood became common. The reason for using narrow strips was for stability, several piece glued together is less likely to warp then a single or pair of boards. Usually when I see strips that have come apart like this it was due to the set being in a damp environment and breaking down the old glue, but at the same time the veneer is usually lifting off.
If there is no warpage, and the joints look straight, and the veneer is not bonded anymore, then I would try gluing the joints back together and clamping them with a set of pipe clamps, perhaps also clamping down vertically with some plastic bag covered blocks and bar clamps or C clamps to keep the joints aligned vertically. If the veneer is still bonded then I don't know what to suggest, this is not something that one typically runs across in my area of the world, the high heat and dry climate must have caused both the wood to shrink and the glue joints to separate.
Regards
Arran