Given a recollement $$ \mathbf{D}^0 \underset{\underset{i_R}\leftarrow}{\overset{\overset{i_L}\leftarrow}\to} \mathbf{D} \underset{\underset{q_R}\leftarrow}{\overset{\overset{q_L}\leftarrow}\to} \mathbf{D}^1 $$ the machinery exposed in BBD's Faisceaux Pervers (Astérisque 100), and in a more modern language in
Banagl, Markus. Topological invariants of stratified spaces. Springer Science & Business Media, 2007.
provides one with a $t$-structure on $\mathbf D$ obtained by "gluing par recollement" a $t$-structure $(\mathcal{D}_\ge^0, \mathcal{D}_\le^0)$ on $\mathbf D_0$ and a $t$-structure $(\mathcal{D}_\ge^1, \mathcal{D}_\le^1)$ on $\mathbf D_1$.
A stable $t$-structure is such that its classes are closed under shifts in both directions.
What happens when I glue two stable $t$-structures par recollement? It seems to me that the result is again stable. Can you provide a reference for this statement, if any, and applications of it to "concrete" situations? Does the theory of stratified spaces, or algebraic perverse sheaves, provide such an application, somewhere?
Note: this question is motivated by the same interest.
