I have now tried for hours and concluded I need someone to show me how it is done. Can anyone show that $p(t, \mathbf{w} | x, \mathbf{x}, \mathbf{t}) = p(t | x, \mathbf{w}) p(\mathbf{w} | \mathbf{x}, \mathbf{t})$?
If relevant, $x$ and $t$ are not dependent on $\mathbf{x}$ and $\mathbf{t}$. You can assume any additional independence if you think it is necessary.
If you assume that $\textbf{w}$ is independent of $x$ and that $t$ is independent of $(\textbf{x}, \textbf{t})$, then you have
$$\begin{align*}p(t,\textbf{w}|x, \textbf{x}, \textbf{t}) &= p(t|\textbf{w}, x, \textbf{x}, \textbf{t})\cdot p(\textbf{w}|x, \textbf{x}, \textbf{t})\\&= p(t|\textbf{w}, x)\cdot p(\textbf{w}|\textbf{x}, \textbf{t}),\end{align*}$$
where the first equality follows from the definition of conditional probability.