Stacks similar to the stack of perverse sheaves

We introduce, on a topological space $X$, a class of stacks of abelian categories we call ``stacks of type P''. This class of stacks includes the stack of perverse sheaves (of any perversity, constructible with respect to a fixed stratification) and is singled out by fairly innocuous axioms. We show that some basic structure theory for perverse sheaves holds for a general stack of type P: such a stack is locally equivalent to a MacPherson-Vilonen construction, and under certain connectedness conditions its category of global objects is equivalent to the category of modules over a finite-dimensional algebra. To prove these results we develop a rudimentary tilting formalism for stacks of type P--another sense in which these stacks are ``similar to stacks of perverse sheaves''.