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[en] Real time evolution of classical gauge fields is relevant for a number of applications in particle physics and cosmology, ranging from the early Universe to dynamics of quark–gluon plasma. We present an explicit non-compact lattice formulation of the interaction between a shift-symmetric field and some gauge sector, , reproducing the continuum limit to order and obeying the following properties: (i) the system is gauge invariant and (ii) shift symmetry is exact on the lattice. For this end we construct a definition of the topological number density that admits a lattice total derivative representation , reproducing to order the continuum expression . If we consider a homogeneous field , the system can be mapped into an Abelian gauge theory with Hamiltonian containing a Chern–Simons term for the gauge fields. This allow us to study in an accompanying paper the real time dynamics of fermion number non-conservation (or chirality breaking) in Abelian gauge theories at finite temperature. When is inhomogeneous, the set of lattice equations of motion do not admit however a simple explicit local solution (while preserving an accuracy). We discuss an iterative scheme allowing to overcome this difficulty.