First Rickard (in Splendid Equivalences: Derived Categories and Permutation Modules ) and then Rouquier (Block theory via stable and Rickard equivalences, Appendix A.1) define splendid equivalences between (principal) blocks of algebras $A$ of $\mathbb{K}G$ and $B$ of $\mathbb{K}H$ with an isomorphic defect group $P$ this way:

$1)$ There exists a complex $X$ of finitely generated $(A,B)$-bimodules such that $\operatorname{Hom}_A (X,X) \simeq B$ and $\operatorname{Hom}_B(X,X) \simeq A$ in the homotopy category of complexes of $B$-modules (resp. $A$-modules), and all terms of $X$ are projective as left and right modules (this is called a split-endomorphism two sided tilting complex)

$2)$ All the terms of $X$, considered as modules of the group algebra of $G \times H$, are relatively projective with respect to the diagonal embedding of $P$ and are $p$-permutation modules (direct summands of permutation modules)

Now, reading both papers, I haven't really been able to understand the requirement that these modules have to be $p$-permutation modules, meaning I don't understand what we lose if we have a split-endomorphism two sided tilting complex made of $\operatorname{diag}(P)$-relatively projective modules that are not $p$-permutation modules.

I am kind of new to this theory, so I'm probably missing something huge. Thanks to anyone who will help me.