Rather than an answer, a comment that is too long to go where it belongs.

I'm guessing that this phenomenon is rarely observed, because mathematicians neither want to nor have much reason to deal with such badly behaved partial orders, since you can construct better behaved order-theoretic structures over them that conserve your ability to reason about the original example.

In recursion theory, mass problems were proposed by Medvedev, a student of Kolmogorov, as a formalisation of Kolmogorov's idea of a "calculus of problems" as a foundation for intuitionistic logic.  They are essentially sets of oracle Turing machines, and so generalise the usual theory of Turing degrees by admitting least upper bounds, and so lattice structure.  Stephen Simpson has done nice work recently, motivated by the idea that there are natural intermediate degrees expressible as mass problems that are not Turing degrees, such as Martin-Löf randomness.  Cf. Simpson (2008) [Mass Problems and Intuitionism][1], Notre Dame Journal of Formal Logic 49(2):127-136.

Recursion theorists are, I think, logicians first and mathematicians second, and so they have a different attitude to formalisation: the structure of Turing degrees is most important to them because that is the basic structure, and that is important enough that they will prefer the sharp edges that come with the first-order formalism.

So if I am not wrong about mathematical culture, I think examples will be hard to find, and may lie behind more widely known structures.

  [1]: www.math.psu.edu/simpson/papers/massint.pdf