= Solution
Suppose $x_0$ has a neighborhood on which $\nabla_\theta\Phi(x,\theta)\ne0$ for every $\theta\ne0$. There one may integrate by parts repeatedly with
$$
L=\frac{\nabla_\theta\Phi}
{i|\nabla_\theta\Phi|^2}\mathbin\cdot\nabla_\theta,
\qquad
Le^{i\Phi}=e^{i\Phi}.
$$
Each adjoint application lowers the effective symbol order. After enough repetitions, the integral and all its $x$-derivatives converge absolutely and define a smooth function near $x_0$. Therefore
$$
\boxed{
\operatorname{sing\,supp}I_\Phi(a)
\subset
\{x:\nabla_\theta\Phi(x,\theta)=0
\text{ for some }\theta\ne0\}}.
$$
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