  A fundamental barrier in extremal hypergraph theory is the presence of many
near-extremal constructions with very different structures. Indeed, the
classical constructions due to Kostochka imply that the notorious extremal
problem for the tetrahedron exhibits this phenomenon assuming Tur\'an's
conjecture.
  Our main result is to construct a finite family of triple systems
$\mathcal{M}$, determine its Tur\'{a}n number, and prove that there are two
near-extremal $\mathcal{M}$-free constructions that are far from each other in
edit-distance. This is the first extremal result for a hypergraph family that
fails to have a corresponding stability theorem.
