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...How does [handheld off-camera flash] work if you're using a 5D3, which has no external flash to trigger the speedlite?
In addition to radio triggers, a very reliable and relatively inexpensive way to do it is to use a cable. For decades we used simple PC cords (sync only), but more recently fully-dedicated cables like the Canon OC-E3 or the Nikon SC-29 have become more common.
I have both short, coiled ETTL cords and a long (24 feet), straight one from Flash Zebra (see photo below), although the latter doesn't get as much use as it used to, now that I have the excellent YN-622C ETTL radio triggers.