= Solution
The condition for a causal linear-filter solution is \b[$|\phi|<1$]. Its mean-square expansion is $X_t=\sum_{j\geq0}\phi^jZ_{t-j}$. Summing the matching <white noise> terms gives
$$
\gamma_X(h)=\frac{\sigma_z^2}{1-\phi^2}\phi^{|h|}.
$$
The assumed orthogonality of every $W_s$ to every $Z_t$ extends to every $X_t$ by L2 convergence of that expansion. Hence the added-noise process has mean zero and
$$
\boxed{\gamma_Y(h)=\frac{\sigma_z^2}{1-\phi^2}\phi^{|h|}+\sigma_w^2\mathbf1_{\{h=0\}}.}
$$
This depends only on lag, proving <weak stationarity>. Its <autocorrelation> has the same geometric tail as the latent autoregression, but its positive-lag correlations are reduced by the additional <variance> at lag zero. This is the <autocovariance of an AR(1) process observed with white noise>. Strict stationarity or Gaussianity is not implied by <white noise> <covariance> assumptions alone.
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