  The Poisson gauge algebra is a semi-classical limit of complete
non-commutative gauge algebra. In the present work we formulate the Poisson
gauge theory which is a dynamical field theoretical model having the Poisson
gauge algebra as a corresponding algebra of gauge symmetries. The proposed
model is designed to investigate the semi-classical features of the full
non-commutative gauge theory with coordinate dependent non-commutativity
$\Theta^{ab}(x)$, especially whose with a non-constant rank. We derive the
expression for the covariant derivative of matter field. The commutator
relation for the covariant derivatives defines the Poisson field strength which
is covariant under the Poisson gauge transformations and reproduces the
standard $U(1)$ field strength in the commutative limit. We derive the
corresponding Bianchi identities. The field equations for the gauge and the
matter fields are obtained from the gauge invariant action. We consider
different examples of linear in coordinates Poisson structures
$\Theta^{ab}(x)$, as well as non-linear ones, and obtain explicit expressions
for all proposed constructions. Our model is unique up to invertible field
redefinitions and coordinate transformations.
