Hint: consider what happens when you make more assumptions about the specific distribution of the errors. Then you can write down exact conditional densities. After multiplying a few together, you will have the joint density of all the time observations, and strong stationarity deals with this joint distribution.
For your model: $$ p(y_1, y_2, \ldots , y_n) = \prod_{t=3}^n p(y_t \mid y_{t-1}, y_{t-2} ) p(y_1, y_2). $$$$ p(y_1, y_2, \ldots , y_n) = \prod_{t=3}^n p(y_t \mid y_{t-1}, y_{t-2} ) p(y_1, y_2)\tag{1}. $$ If you assumed that the errors were Normally distributed then $$ p(y_t \mid y_{t-1}, y_{t-2} ) = N(.8 y_{t-1} +.1 y_{t-2}, \sigma^2). $$
Another hint: If this Normal distribution does lead to strong stationarity, then the joint distribution of all the observations $\{y_t\}$ should have the right means, and the right variances and (auto-)covariances. Arrange all of those autocovariances and variances into a matrix $\Gamma$. Then your joint density should be $$ (2\pi)^{-n/2}(\det\Gamma)^{-1/2}\exp\left[-\frac{1}{2}\mathbf{y}_t'\Gamma^{-1}\mathbf{y}_t \right]. $$