TITLE: Counterexamples to gluing complexes of sheaves
QUESTION [13 upvotes]: Note: I asked the question below last week on MathSE but received no answer. 
Background:
I have read the claim that perverse sheaves behave more like sheaves than like complexes of sheaves.  This refers to the fact that they can be glued.  
For instance, suppose that $X$ is a complex analytic space and $P_1^{\bullet}, P_2^{\bullet}$ are perverse sheaves defined on the open sets $U_1, U_2$ respectively. Then if we are given an isomorphism $\alpha_{12}: P_1^{\bullet}|_{U_1 \cap U_2} = P_2^{\bullet}|_{U_1 \cap U_2},$ then there exists a unique (up to canonical isomorphism) perverse sheaf $P^{\bullet}$ defined on $U_1 \cup U_2$ with the property that there are isomorphisms $\beta_i:P^{\bullet}|_{U_i} \to P_i^{\bullet}$ with ${\beta_2}|_{U_1 \cap U_2}={(\alpha_{12} \circ \beta_1) }|_{U_1 \cap U_2}.$
More generally, if one has an open cover $\{U_i\}$ of $X$ and perverse sheaves $P_i^{\bullet}$ of with isomorphisms on the overlaps $U_i \cap U_j$ satisfying the co-cycle condition, then this data glue in the usual way. 
My question: 
What goes wrong if one tries to glue ordinary complexes of sheaves?  Are there counterexamples showing that the gluing property cannot hold in $C^{\bullet}(Sh(X))$ (the category of sheaves on $X$) or in $D^b(X)$ (the bounded derived category of sheaves on $X$)?

REPLY [3 votes]: I'm not sure if you are still interested in this question. Actually for an open cover $\{U_i\}$ and complexes of sheaves on each $U_i$, we could give the "higher" descent data and "higher" cocycle conditions in terms of twisted complexes, which was first introduced by O’Brian, Toledo and Tong in late 1970's and 1980's. I have written a paper to describe twisted complexes as a dg-enhancement of the derived category of sheaves on the space (sorry for the self-citation). Later in a joint paper with Block and Holstein we proved that twisted complexes indeed give the homotopy limit of cosimplicial diagrams of dg-categories coming from geometry (we can consider homotopy limit as a more formal way to say descent data).
I would like to mention that in a recent preprint, the authors generalized the main result in the paper of Block, Holstein and me to arbitrary cosimplicial diagrams of dg-categories.