This doesn't need to hold. For example, if one takes a $(T^4,\omega)$ with a constant symplectic structure $\omega$, in order for it to have a fibration by Lagrangian tori one should be able to find a homologically non-trivial $T^2\subset T^4$ such that $\int_{\omega} T^2=0$ which is impossible for general $\omega$.
One can also give counter-examples when a half-dimensional torus bundle exists on the symplectic manifold but no Lagrangian torus bundle exists for any symplectic structure on the manifold. To construct such an example consider $M^4$ that fibers over $B=T^2$ with a fiber $F=T^2$ such that the action of $\mathbb Z^2=\pi_1(B)$ on $H_1(F)=\mathbb Z^2$ contains a hyperbolic element. Such a manifold is symplectic by Thurston, but it is not a total space of a Lagrangian torus fibration by the classification of such fibrations in dimension $4$.