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?

Note: this question is motivated by the same interest.

fourdistinct (but "compatible" in a sense) recollements. I can expand on this, if it's what you mean $\endgroup$6more comments