= Solution
Stack $y=(y_1^\top,y_2^\top)^\top$. After integrating out the <Ornstein-Uhlenbeck process>,
$$
y\mid t,\theta\sim N_{2N}(\mu_\theta,\Sigma_\theta),
\qquad
\mu_\theta=
\begin{pmatrix}c\mathbf1\\(c+\Delta m)\mathbf1\end{pmatrix}.
$$
Writing $k(a,b)=A^2e^{-|a-b|/\tau}$, the covariance blocks are
$$
(\Sigma_{11})_{jk}=k(t_j,t_k)+\sigma_{1,j}^2\mathbf1_{\{j=k\}},
$$
$$
(\Sigma_{22})_{jk}=k(t_j-\Delta t,t_k-\Delta t)+\sigma_{2,j}^2\mathbf1_{\{j=k\}},
$$
and $(\Sigma_{12})_{jk}=k(t_j,t_k-\Delta t)$, with $\Sigma_{21}=\Sigma_{12}^\top$. Therefore
$$
p(y_1,y_2\mid t,\theta)
=(2\pi)^{-N}|\Sigma_\theta|^{-1/2}
\exp\left[-\frac12(y-\mu_\theta)^\top
\Sigma_\theta^{-1}(y-\mu_\theta)\right].
$$
Solved by gpt-5.6-sol high.
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