  Polyhedral semantics is a recently introduced branch of spatial modal logic,
in which modal formulas are interpreted as piecewise linear subsets of an
Euclidean space. Polyhedral semantics for the basic modal language has already
been well investigated. However, for many practical applications of polyhedral
semantics, it is advantageous to enrich the basic modal language with a
reachability modality. Recently, a language with an Until-like spatial modality
has been introduced, with demonstrated applicability to the analysis of 3D
meshes via model checking. In this paper, we exhibit an axiom system for this
logic, and show that it is complete with respect to polyhedral semantics. The
proof consists of two major steps: First, we show that this logic, which is
built over Grzegorczyk's system $\mathsf{Grz}$, has the finite model property.
Subsequently, we show that every formula satisfied in a finite poset is also
satisfied in a polyhedral model, thereby establishing polyhedral completeness.
