  Inspired by the work of Pomoni-Yan-Zhang in String Theory, we introduce the
moduli space of tetrahedron instantons as a Quot scheme and describe it as a
moduli space of quiver representations. We construct a virtual fundamental
class and virtual structure sheaf \`a la Oh-Thomas, by which we define
K-theoretic invariants. We show that the partition function of such invariants
reproduces the one studied by Pomoni-Yan-Zhang, and explicitly determine it, as
a product of shifted partition functions of rank one Donaldson-Thomas
invariants of the three-dimensional affine space. Our geometric construction
answers a series of questions of Pomoni-Yan-Zhang on the geometry of the moduli
space of tetrahedron instantons and the behaviour of its partition function,
and provides a new application of the recent work of Oh-Thomas.
