Merging lists of fragile objects
No, such an algorithm cannot exist.

Assume $t$ comparisons per element are allowed.

For a start, consider the situation of merging two lists, one of size
one, and the other of size $2^t$.
There are $2^t+1$ possible results, and an easy adversarial argument
shows that the element in the small list needs to participate in $t+1$ comparisons,
and this element has to go to the trash.

From this it is easy to construct an instance with two lists of size $n$
that consists of $n/2^t$ such small situations, and hence $n/2^t$ elements
have to go to the trash.

Hence, as long as $t$ is constant, it is impossible to have a trash of
$o(n)$.

On a side note, it seems that it is possible to match this bound by an algorithm where every element is compared with roughly log of the size of the surrounding part of the other list.  If this is of interest, I will try to work out the details.