sg.symplectic geometry - To what extent can I think of a Lagrangian fibration in a symplectic manifold as T*N?

Dear Theo Johnson-Freyd, I hope to have at least partially understood the content of your question, and that my answer could be useful.

0.Setting and specification of the terminology.
In a symplectic $2n$-dimensional manifold $(M,omega)$, let be given a lagrangian foliation $mathcal{F}$, i.e. a foliation of $M$ whose leaves are lagrangian w.r.t. $omega$.
(Instead, I mean a lagrangian fibration of $(M,omega)$ as a surjective summersion $f:Mto B$ whose fibers are lagrangian w.r.t. $omega$. Any fibration determines a foliation but the converse is not true. The difference will be immaterial in my point(1), but not so in my point(2).)

1.Local Existence of lagrangian submanifolds transversal to $mathcal{F}$.
For any $pin M$, there exists a lagrangian submanifold of $(M,omega)$ which passes through $p$ and is transversal to $mathcal{F}$.

Infact, for any $pin M$, there exists a chart $(U,phi)$ for $M$ centered at $p$, such that:
$omega= sum_{i=1}^{n}{dphi_i wedge dphi_{n+i}}$,
the restriction of $mathcal{F}$ on $U$ is generated by $frac{partial}{partialphi_{n+1}},ldots,frac{partial}{partialphi_{2n}}$,
and consequently $phi_{n+1}=ldots=phi_{2n}=0$ is a local lagrangian submanifold of $(M,omega)$ passing through $p$ and transverval to $mathcal{F}$.

This is just the Caratheodory-Jacobi-Lie theorem, applied starting with a system $dphi_1,ldots,dphi_n$ of $1$-forms which locally generates the distribution corresponding to the lagrangian foliation $mathcal{F}$.

2. A relative globalization.
If $L$, a lagrangian submanifold of $(M,omega)$, is transversal to $mathcal{F}$, then there exists a diffeomorphism $f$ from an open neigborhood of $L$ in $M$ onto an open set in $T^*L$ such that:
$f|_L$ is the zero section of $tau_L^{ast}:T^{ast}Lto L$,
$f_{ast}omega$ is the canonical symplectic on $T^{ast}L$,
and $f$ takes the leaves of $mathcal{F}$ in the fibers of $tau^{ast}_L$.

This is just Theorem 7.1 in "Symplectic Manifolds and their Lagrangian submanifolds" of A.Weinstein.