Task: Given an undirected graph $G = (V, E)$, find a largest matching $M \subseteq E$ such that $G-M$ has no Eulerian components (i.e. all connected components of $G-M$ must have odd-degree vertices).
Of course, $G$ can be assumed to be connected, otherwise one could solve the problem for each connected component individually. If $G$ is Eulerian, any maximum matching would solve the problem: In this case, $G - M$ has only non-Eulerian components because at least one matching edge must end in every component (thanks for Perry for pointing this out).
So the interesting case is when $G$ is connected and not Eulerian. In this case, $M = \varnothing$ would be one matching with the required property, so there exists a solution. I neither found an efficient method to solve this problem nor did I find a proof that it is $NP$-hard. Moreover, I didn't find any previous work discussing it.
Also, what about special cases? For example, what if $G$ is $d$-regular? If $d$ is odd, every perfect matching would not have the required property. A near-perfect matching would have the property if and only if $G - M$ is still connected.
Motivation: I came across this problem when studying an application in VLSI design. In particular, a very specific problem of arranging transistors in a row within the minimum amount of space boils down to this graph-theoretic question.