  We study operators of the form $$ Tf(x)= \psi(x) \int f(\gamma_t(x))K(t)\,dt,
$$ where $\gamma_t(x)$ is a real analytic function of $(t,x)$ mapping from a
neighborhood of $(0,0)$ in $\mathbb{R}^N \times \mathbb{R}^n$ into
$\mathbb{R}^n$ satisfying $\gamma_0(x)\equiv x$, $\psi(x) \in
C_c^\infty(\mathbb{R}^n)$, and $K(t)$ is a "multi-parameter singular kernel"
with compact support in $\mathbb{R}^N$; for example when $K(t)$ is a product
singular kernel. The celebrated work of Christ, Nagel, Stein, and Wainger
studied such operators with smooth $\gamma_t(x)$, in the single-parameter case
when $K(t)$ is a Calder\'on-Zygmund kernel. Street and Stein generalized their
work to the multi-parameter case, and gave sufficient conditions for the
$L^p$-boundedness of such operators. This paper shows that when $\gamma_t(x)$
is real analytic, the sufficient conditions of Street and Stein are also
necessary for the $L^p$-boundedness of $T$, for all such kernels $K$.
