Suppose $S$ is a symmetric monoidal groupoid. Take Quillen and Grayson's $S^{-1}S$-construction, which is a symmetric monoidal category with objects given by pairs $(m,n)$ and maps given by compositions of "formal additions" $s+:(m,n) \rightarrow (s+m,s+n)$ and pairs of morphisms in $S$. There is an endofunctor $\iota : S \rightarrow S$ that switches $(m,n)$ to $(n,m)$.
In the paper "Beware the Phony Multiplication on Quillen's $\mathscr{a}^{-1}\mathscr{a}$" Thomason argued that there cannot be a natural transformation
$$(0,0) \rightarrow (m+n,n+m).$$
In other words, there is no natural transformation that verifies that $\iota$ is an inverse before realization. In section IV.4 in Weibel's K-Book, Weibel gives this as an exercise. However, he also claims that Thomason has a proof that $B(\iota)$ is in fact the homotopy inverse on $B(S^{-1}S)$, but for subtle reasons. Does anyone know this argument or a reference?
