  In this paper we refine and extend the results of arXiv:1701.04138, where a
connection between the $AdS_{5}\times S^{5}$ superstring lambda model on
$S^{1}=\partial D$ and a double Chern-Simons (CS) theory on $D$ based on the
Lie superalgebra $\mathfrak{psu}(2,2|4)$ was suggested, after introduction of
the spectral parameter $z$. The relation between both theories mimics the
well-known CS/WZW symplectic reduction equivalence but is non-chiral in nature.
All the statements are now valid in the strong sense, i.e. valid on the whole
phase space, making the connection between both theories precise. By
constructing a $z$-dependent gauge field in the 2+1 Hamiltonian CS theory it is
shown that: i) by performing a symplectic reduction of the CS theory the
Maillet algebra satisfied by the extended Lax connection of the lambda model
emerges as a boundary current algebra and ii) the Poisson algebra of the
supertraces of $z$-dependent Wilson loops in the CS theory obey some sort of
spectral parameter generalization of the Goldman bracket. The latter algebra is
interpreted as the precursor of the (ambiguous) lambda model monodromy matrix
Poisson algebra prior to the symplectic reduction. As a consequence, the
problematic non-ultralocality of lambda models is avoided (for any value of the
deformation parameter $\lambda \subset [0,1]$), showing how the lambda model
classical integrable structure can be understood as a byproduct of the
symplectic reduction process of the $z$-dependent CS theory.
