  Cluster algebras *with coefficients* are important since they appear in
nature as coordinate algebras of varieties like Grassmannians, double Bruhat
cells, unipotent cells, ... . The approach of Geiss-Leclerc-Schr\"oer often
yields Frobenius exact categories which allow to categorify such cluster
algebras. In previous work, the third-named author has constructed Higgs
categories and relative cluster categories in the relative Jacobi-finite
setting (arXiv:2109.03707). Higgs categories generalize the Frobenius
categories used by Geiss-Leclerc-Schr\"oer.
  In this article, we construct the Higgs category and the relative cluster
category in the relative Jacobi-infinite setting under suitable hypotheses.
These cover for example the case of Jensen-King-Su's Grassmannian cluster
category. As in the relative Jacobi-finite case, the Higgs category is no
longer exact but still extriangulated in the sense of Nakaoka-Palu. We also
construct a cluster character refining Plamondon's.
  In the appendix, Chris Fraser and the second-named author categorify
quasi-cluster morphisms using Frobenius categories. A recent application of
this result is due to Matthew Pressland, who uses it to prove a conjecture by
Muller-Speyer.
