Obstructions to homotopy invariance of loop coproduct via parametrised fixed-point theory

Given f: M → N a homotopy equivalence of compact manifolds with boundary, we use a construction of Geoghegan and Nicas to define its Reidemeister trace [T] ∈ π₁ˢᵗ(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.