Spectral coverings without embeddings

In this article, we investigate a weakened version of the spectral correspondence for twisted Higgs bundles. Namely, we construct twisted Higgs bundles from a finite covering map and a vector bundle on that covering but without requiring that they match the eigen-data for some fixed twisted Higgs bundle. We investigate stability for twisted Higgs bundles constructed in this way, and compare our covering data to that of the traditional spectral cover.