  The Poisson gauge theory is a semi-classical limit of full non-commutative
gauge theory. In this work we construct an L$_\infty^{full}$ algebra which
governs both the action of gauge symmetries and the dynamics of the Poisson
gauge theory. We derive the minimal set of non-vanishing $\ell$-brackets and
prove that they satisfy the corresponding homotopy relations. On the one hand,
it provides new explicit non-trivial examples of L$_\infty$ algebras. On the
other hand, it can be used as a starting point for bootstrapping the full
non-commutative gauge theory. The first few brackets of such a theory are
constructed explicitly in the text. In addition we show that the derivation
properties of $\ell$-brackets on L$_\infty^{full}$ with respect to the
truncated product on the exterior algebra are satisfied only for the canonical
non-commutativity. In general, L$_\infty^{full}$ does not have a structure of
P$_\infty$ algebra.
