noncommutative geometry - Gluing perverse sheaves?

Ryan Reich has a paper called Notes on Beilinson's "How to glue perverse sheaves" -- it's also available off of Dennis Gaitsgory's Geometric Representation Theory page which is an amazing resource in the area. The paper by MacPherson and Vilonen, Elementary construction of perverse sheaves. Invent. Math. 84 (1986), no. 2, 403--435
provides another perspective on the same construction of perverse sheaves by gluing.

I don't know of a way to construct perverse sheaves in any noncommutative generality, and it seems like an unlikely proposition to me. You can certainly define modules over the de Rham complex and the like in great generality, but the condition of constructibility (not to mention the perverse t-structure) seems very commutative, involving restricting supports and dimensions etc. You can define holonomicity homologically, so once you have a notion of D-modules you can almost reverse engineer perverse sheaves (not sure how you'd describe regularity truly noncommutatively), but I don't see how you'd find a non-definitional version of the Riemann-Hilbert correspondence unless you're secretly in an almost commutative setting. (For that matter D-modules themselves are an almost-commutative notion..)