  We consider homological mirror symmetry in the context of hypertoric
varieties, showing that appropriate categories of B-branes (that is, coherent
sheaves) on an additive hypertoric variety match a category of A-branes on a
Dolbeault hypertoric manifold for the same underlying combinatorial data. For
technical reasons, the category of A-branes we consider is the modules over a
deformation quantization (that is, DQ-modules). We consider objects in this
category equipped with an analogue of a Hodge structure, which corresponds to a
$\mathbb{G}_m$-action on the dual side of the mirror symmetry.
  This result is based on hands-on calculations in both categories. We analyze
coherent sheaves by constructing a tilting generator, using the characteristic
$p$ approach of Kaledin; the result is a sum of line bundles, which can be
described using a simple combinatorial rule. The endomorphism algebra $H$ of
this tilting generator has a simple quadratic presentation in the grading
induced by $\mathbb{G}_m$-equivariance. In fact, we can confirm it is Koszul,
and compute its Koszul dual $H^!$.
  We then show that this same algebra appears as an Ext-algebra of simple
A-branes in a Dolbeault hypertoric manifold. The $\mathbb{G}_m$-equivariant
grading on coherent sheaves matches a Hodge grading in this category.
