= Solution
For the proposed <joint probability density> $\rho$, define the acceptance <indicator function>
$$
A_i:=\mathbf1_{\{V_i\leq w(Y_i)/M\}}.
$$
On the proposal's <support> the <importance weight> is $\pi(Y_i,V_i)/\rho(Y_i,V_i)=MA_i$. Therefore the requested augmented-space <importance sampling> estimator is
$$
\boxed{\widehat\theta_n^{\mathrm{IS}}=\frac Mn\sum_{i=1}^nA_i\phi(Y_i).}
$$
Its <expected value> is $\theta$, provided $\mathbb E_p|\phi(X)|<\infty$. This is not the ordinary estimator $n^{-1}\sum_iw(Y_i)\phi(Y_i)$ from question 1. Taking the <conditional expectation> of this augmented estimator given all $Y_i$ produces that ordinary estimator, because $\mathbb E[A_i\mid Y_i]=w(Y_i)/M$. This is the <Rao-Blackwell identity for a fixed-denominator rejection estimator>.
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