Let $(\Omega, \mathcal A, \mathbb P)$ be a probability space and $X=(X_t)$ a real Gaussian stochastic process.
Let $\mathcal F=(\mathcal F_t)$ be the filtration generated by $(X_t)$.
$X$ is Markov with regard to $\mathcal F$ iff $\mathbb P [X_t \in \cdot | \mathcal F_s ] = \mathbb P [X_t \in \cdot | X_s ]$ for $s<t$.
$X$ is semi-Markov with regard to $\mathcal F$ iff there exists an $n>0$ such that $\mathbb P [X_t \in \cdot | \mathcal F_s ] = \mathbb P [X_t \in \cdot | X_{u_1}, \cdots, X_{u_n}, X_s ]$ for $s<t$ and any $u_1<\ldots < u_n<s$. Here I excluded Markov processes from the class of semi-Markov processes.
Brownian motion is an example of Markov process. What is a canonical example of a Gaussian semi-Markov process ?