  We study four (a priori) different ways in which an open book decomposition
of the 3-sphere can be defined to be braided. These include generalised
exchangeability defined by Morton and Rampichini and mutual braiding defined by
Rudolph, which were shown to be equivalent by Rampichini, as well as
P-fiberedness and a property related to simple branched covers of $S^3$
inspired by work of Montesinos and Morton. We prove that these four notions of
a braided open book are actually all equivalent to each other. We show that all
open books in the 3-sphere whose binding has a braid index of at most 3 can be
braided in this sense.
