  Given $f: M \to N$ a homotopy equivalence of compact manifolds with boundary,
we use a construction of Geoghegan and Nicas to define its Reidemeister trace
$[T] \in \pi_1^{st}(\mathcal{L} N, N)$. We realize the Goresky-Hingston
coproduct as a map of spectra, and show that the failure of $f$ to entwine the
spectral coproducts can be characterized by Chas-Sullivan multiplication with
$[T]$. In particular, when $f$ is a simple homotopy equivalence, the spectral
coproducts of $M$ and $N$ agree.
