Seeking shapefiles of global climate/climatic zones
A respected source of global climate classification information can be found here: http://koeppen-geiger.vu-wien.ac.at/present.htm. The data are in KMZ rather than shapefile format, but should be easy to convert. The value of using these is that the methodology for their creation is well documented (see citations here).
I have generated a shapefile from the raster mentioned above. I simply used the "Raster to Vector" tool in QGIS.
You may decide to create the shapefile by yourself or simply download it from my website here: https://geoafrikana.com/resources/