I know that given two problems are undecidable it does not follow that their intersection must be undecidable. For example, take a property of languages $P$ such that it is undecidable whether the language accepted by a given pushdown automaton $M$ has that property. Clearly $P$ and $\lnot P$ are undecidable (for a given $M$) but $P \cap \lnot P$ is trivially decidable (it is always false).I wonder if there are any real life examples which do not make use of the trick above? When I say real life I do not necessarily mean problems which people come across in their day to day life, I mean examples where we do not take a problem and it's complement. It would be interesting (to me) if there are examples where the intersection is not trivially decidable.
Examples of undecidable problems whose intersection is decidable
reference request;undecidability;decision problem
So here is a example, which is probably not as nice as you wanted it to be, but less trivial than the one you have mentioned.Let $L_1,L_2\subset \{a,b,c\}^*$ be two undecidable languages, and $L_3\subseteq \{a,b,c\}^*$ a decidable language. We define\begin{align}L_A&:=\{a\,w \mid w\in L_1\} \cup \{c\,w \mid w\in L_3\}, \\ L_B&:=\{b\,w \mid w\in L_2\} \cup \{c\,w \mid w\in L_3\} .\\   \end{align}Clearly, both $L_A$ and $L_B$ are not decidable, however their nonempty intersection$$ L_A\cap L_B =\{c\,w \mid w\in L_3\}$$is decidable.