Physically, we know that a BEC has formed if a macroscopic number of bosons occupy a single quantum state. The wave-function $\Psi(x)$ of the latter, normalized to the total number of condensed atoms $N \gg 1$, gives the macroscopic description of the condensate. $\Psi(x)$ will satisfy the ususal Schrodinger euqation in the trapping potential (which is the special case of the Gross-Pitaevskii equation with zero interactions).
On the other had, a macroscopic limit of a boson field should be described by the (classical) Klein-Gordon equation. In the relevant non-realtivistic limit, dispersion of the Klein-Gordon field is quadratic, as in the Schrodinger equation, but I struggle to derive the former from the latter.
Despite this technical difficulty of mine, is it a valid claim that BEC is a physical realization of the non-relativistic, classical-wave-limit of a boson field? I see it in the same vein as electric-field description of the laser radiation being a classic limit of the quantum coherent state.
(The difference between number states and coherents states in the limit $N \gg 1$ is, hopefully, a mere technicality here).