During some self studies of time series and ARCH processes, I thought of the following example.
Given an $\mathrm{ARCH}(1)$ process $Z_t$, \begin{align*} Z_t &= e_t\sqrt{h_t}\\ h_t&=\alpha_0+\alpha_1Z_{t-1}^2 \end{align*} where $e_t$ is a sequence of IID standard Gaussian rvs and $\alpha_0\geq0$ and $\alpha_1\in[0,1)$. By Brockwell and Davis (2016) $Z_t$ is strictly stationary.
Next, suppose we let $X_t$ be a causal $\mathrm{ARMA}(1,1)$ process $\phi(B)X_t=\theta(B)Z_t$. Then clearly $X_t$ is weakly stationary. However, is it also strictly stationary? Spontaneously, I'd think that strict stationarity of $X_t$ would follow by strict stationarity of $Z_t$ but have no clue of how one would prove that. My searches online on the topic have not yielded much of an answer.
Your intuition is correct, $X_t$ is strictly stationary. Here's a slightly more general theorem that you can use to show this.
Theorem Let $(\epsilon_t)_{t \in \mathbb{Z}}$ be a strictly stationary sequence and $(\psi_t)_{t \in \mathbb{Z}}$ an absolutely summable sequence of real numbers. Define the causal linear process $$X_t = \sum_{k=1}^\infty \psi_k \epsilon_{t-k}$$ Then, the process $X_t$ is strictly stationary.
Proof. Note that for any $h \in \mathbb{Z}$, $(\epsilon_t)_{t \in \mathbb{Z}}$ and $(\epsilon_{t+h})_{t \in \mathbb{Z}}$ are two processes with the same finite-dimensional distributions (by the strict stationarity of $\epsilon_t$). Hence, $(\epsilon_t)_{t \in \mathbb{Z}} \stackrel{D}{=} (\epsilon_{t+h})_{t \in \mathbb{Z}}$.
Now, let $\Psi : \mathbb{R}^{\mathbb{Z}} \to \mathbb{R}^{\mathbb{Z}}$ be given by $$\Psi ((e_t)_{t \in \mathbb{Z}}) = (x_t)_{t \in \mathbb{Z}}$$ where $x_t = \sum_{k=1}^\infty \psi_k e_{t-k}$. This map is measurable in the product $\sigma$-algebra $\bigotimes_{k \in \mathbb{Z}} \mathcal{B}(\mathbb{R})$. Thus, it follows that the stochastic processes $\Psi ((\epsilon_t)_{t \in \mathbb{Z}})$ and $\Psi ((\epsilon_{t+h})_{t \in \mathbb{Z}})$ agree in distribution, which is precisely the statement that $X_t$ is strictly stationary. $\blacksquare$
The advantage of using this theorem is that you can apply it to more general $\mathrm{ARMA-GARCH}$ sequences, provided that the $\mathrm{GARCH}$ innovations form a strictly stationary sequence and that the $\mathrm{ARMA}$ equations admit a causal representation.