To date, both sides in the conventionality of simultaneity debate grant that transformations from "standard" to "nonstandard" coordinates are possible without any empirically significant effects. However, it is argued here that the very possibility of defining nonstandard coordinates vanishes if one represents special relativity, not by real four-vectors (as has been the case so far in the debate), but by complex spinors as used in the representation of half-integer spin. Thus, in the topologically simplest representation of the Lorentz group, the (...) transformation over which the debate has been fought does not exist and one of the most basic foundations of the conventionalist thesis is challenged. (shrink)