  In this paper, we realize polynomial $\H$-modules
$\Omega(\lambda,\alpha,\beta)$ from irreducible twisted Heisenberg-Virasoro
modules $\A_{\alpha,\beta}$. It follows from $\H$-modules
$\Omega(\lambda,\alpha,\beta)$ and $\mathrm{Ind}(M)$ that we obtain a class of
natural non-weight tensor product modules
$\big(\bigotimes_{i=1}^m\Omega(\lambda_i,\alpha_i,\beta_i)\big)\otimes
\mathrm{Ind}(M)$. Then we give the necessary and sufficient conditions under
which these modules are irreducible and isomorphic, and also give that the
irreducible modules in this class are new.
