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 $(\mathcal{D}_\ge^0, \mathcal{D}_<^0)$ on $\mathbf D^0$ and a $t$-structure $(\mathcal{D}_\ge^1, \mathcal{D}_<^1)$ on $\mathbf D^1$.
Edit: Apparently this construction deserves a more detailed explanation. When you have a t-structure $\mathcal{D}^i_{\ge}$ (identified with its left aisle) on $\mathbf{D}^i$ you can define the gluing $$ (\mathcal{D}^0_\ge\wr\mathcal{D}_\ge^1, \mathcal{D}^0_<\wr\mathcal{D}_<^1) $$ exploiting the functors $(i_L\dashv i\dashv i_R), (q_L\dashv q\dashv q_R)$: define $$\begin{gather} \{ X\in\mathbf{D}\mid qX\in \mathcal{D}_\ge^1, \; i_L X \in\mathcal{D}_\ge^0\}\\ \{ Y\in\mathbf{D}\mid qY \in\mathcal{D}_<^1,\; i_R Y \in\mathcal{D}_<^0 \} \end{gather}$$ It can be shown that these two classes form a new t-structure.
Now, [BBD] and Banagl book, starting from the classical geometric example of a stratification $\varnothing\subset U\subset X$ for a topological space, both insist on the fact that
By applying the gluing Theorem [i.e. the construction providing the glued $t$-structure] inductively, the notion of a p-perverse t-structure quickly generalizes to spaces with more than two strata
The possibility of doing this ultimately relies into the fact that gluing par recollement is an associative operation, namely (in some sense to be specified) if we denote the [left aisle of the] glued $t$-structure above as $\mathcal{D}^0_\ge\wr\mathcal{D}_\ge^1$, we have $$ (\mathcal{D}^0_\ge\wr\mathcal{D}_\ge^1)\wr \mathcal{D}^2_\ge = \mathcal{D}^0_\ge\wr(\mathcal{D}_\ge^1\wr \mathcal{D}^2_\ge). $$ In the geometric setting, I expect this result ultimately depend on the commutation of some diagrams of adjoint functors. Namely, the diagram
is commutative (in an obvious sense), where a squiggly arrow $f\colon \mathbf{D}\to \mathbf{D}'$ denotes a triple of adjoints $(f_L\dashv f\dashv f_R)$. In the geometric case, these maps are direct/inverse images, and the presence of a stratification $\varnothing U\subset V\subset X$ gives the desired compatibility.
Now,
What happens in the general setting of an "abstract" recollement in a generic triangulated (or better, stable) category? Is there any property of recollements ensuring that the glued $t$-structure $\mathcal{D}_0 \wr\dots \wr \mathcal{D}_n$ exists in a definite sense?
