The mathematicians' idiom for a wide class of situations resembling this one is "blow-up", inspired by the term from algebraic geometry. It is used both as noun (a blow up of ...) and verb (to blow up ...).
It means to start from one structure and blow up (inflate) part of it, forming a larger or more complicated structure, keeping other parts the same.
For this example of "exact pairs", one can construct the partial order $P$ by starting with linear order consisting of the chain $a_i$ and its least upper bound $L$, then blowing up $L$ into a pair of points $b,c$. Identifying $b$ and $c$ is a quotient of posets that reverses the blow up (ie, restores $L$ in its original state of being a least upper bound to the $a_i$) and this is equivalent to the definition of exact pair. It is also clear from this observation how to define exact triples, or splittings of several least upper bounds.
So outside of recursion theory, I think math people would commonly describe such a diagram as a (two-fold) splitting or blowup of a point in a poset. I don't know of any compelling examples where this construction occurs but certainly the idea would feel very familiar to many if phrased in the language of blow-ups.
EDIT: something close to what Joel is asking about is the theory of R-trees (R as in "real numbers"). The minimal example of a dense order where any chain has a least exact-pair of upper bounds is some sort of infinite trivalent R-tree.