  We develop a Sz.-Nagy--Foias-type functional model for a commutative
contractive operator tuple $\underline{T} = (T_1, \dots, T_d)$ having $T = T_1
\cdots T_d$ equal to a completely nonunitary contraction. We identify
additional invariants ${\mathbb G}_\sharp, {\mathbb W}_\sharp$ in addition to
the Sz.-Nagy--Foias characteristic function $\Theta_T$ for the product operator
$T$ so that the combined triple $({\mathbb G}_\sharp, {\mathbb W}_\sharp,
\Theta_T)$ becomes a complete unitary invariant for the original operator tuple
$\underline{T}$. For the case $d \ge 3$ in general there is no commutative
isometric lift of $\underline{T}$; however there is a (not necessarily
commutative) isometric lift having some additional structure so that, when
compressed to the minimal isometric-lift space for the product operator $T$,
generates a special kind of lift of $\underline{T}$, herein called a {\em
pseudo-commutative contractive lift} of $\underline{T}$, which in turn leads to
the functional model for $\underline{T}$. This work has many parallels with
recently developed model theories for symmetrized-bidisk contractions
(commutative operator pairs $(S,P)$ having the symmetrized bidisk $\Gamma$ as a
spectral set) and for tetrablock contractions (commutative operator triples
$(A, B, P)$ having the tetrablock domain ${\mathbb E}$ as a spectral set).
