= Solution
For a <Sierpiński decomposition of the plane>, $A,B$ partition $\mathbb R^2$, and one has countable vertical sections while the other has countable horizontal sections. With the orientation fixed as follows,
$$
\boxed{A\cap B=\varnothing,\quad A\cup B=\mathbb R^2,\quad
|\{y:(x,y)\in A\}|\leq\aleph_0,\quad
|\{x:(x,y)\in B\}|\leq\aleph_0}
$$
for every real $x,y$. Interchanging the coordinate directions and the names $A,B$ gives the equivalent convention. Here “countable” includes finite sets.
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