  For every positive integer $t$ we construct a finite family of triple systems
${\mathcal M}_t$, determine its Tur\'{a}n number, and show that there are $t$
extremal ${\mathcal M}_t$-free configurations that are far from each other in
edit-distance. We also prove a strong stability theorem: every ${\mathcal
M}_t$-free triple system whose size is close to the maximum size is a subgraph
of one of these $t$ extremal configurations after removing a small proportion
of vertices. This is the first stability theorem for a hypergraph problem with
an arbitrary (finite) number of extremal configurations. Moreover, the extremal
hypergraphs have very different shadow sizes (unlike the case of the famous
Tur\'an tetrahedron conjecture). Hence a corollary of our main result is that
the boundary of the feasible region of ${\mathcal M}_t$ has exactly $t$ global
maxima.
