= Solution
Use $\mathbb C^2$ with the <standard symplectic form> $\omega_0=dx_1\wedge dy_1+dx_2\wedge dy_2$ and the <compatible almost complex structure> given by multiplication by $i$. Consider the <embedded submanifold>
$$
\boxed{L=\{(s+it,t):s,t\in\mathbb R\}.}
$$
In coordinates $(x_1,y_1,x_2,y_2)$, its <tangent space> is $W=\{(a,b,b,0):a,b\in\mathbb R\}$. Applying $J$ gives $J(a,b,b,0)=(-b,a,0,b)$. If this is again in $W$, its fourth coordinate forces $b=0$, and its second and third coordinates then force $a=0$. Thus $W\cap JW=0$ and $\dim_\mathbb R L=2$: $L$ is a <totally real submanifold>.
However, the <pullback of a differential form> under the parametrization is
$$
\boxed{\omega_0|_L=ds\wedge dt\ne0.}
$$
Thus \b[this <totally real submanifold> is not a <Lagrangian submanifold>].
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