  In earlier work, the author introduced a method for constructing a Frobenius
categorification of a cluster algebra with frozen variables by starting from
the data of an internally Calabi-Yau algebra, which becomes the endomorphism
algebra of a cluster-tilting object in the resulting category. In this paper,
we construct appropriate internally Calabi-Yau algebras for cluster algebras
with polarised principal coefficients (which differ from those with principal
coefficients by the addition of more frozen variables), and obtain Frobenius
categorifications in the acyclic case. Via partial stabilisation, we then
define extriangulated categories, in the sense of Nakaoka and Palu,
categorifying acyclic principal coefficient cluster algebras, for which
Frobenius categorifications do not exist in general. Many of the intermediate
results used to obtain these categorifications remain valid without the
acyclicity assumption, as we will indicate, and are interesting in their own
right. Most notably, we provide a Frobenius version of Van den Bergh's result
that the Ginzburg dg-algebra of a quiver with potential is bimodule
3-Calabi-Yau.
