The notion of right Bousfield delocalization reverses this relation: $M_1$ is a right Bousfield delocalization of $M_2$ if $M_2$ is a right Bousfield localization of $M_1$.

Of course, the nontrivial task here is to establish interesting existence criteria for right Bousfield delocalizations.

Existence theorem

Theorem (Corrigan-Salter)

If $M_1$ and $M_2$ are two cofibrantly generated model category structures on the same category with coinciding classes of fibrations, then there is a third cofibrantly generated model structure $M_3$ with the same fibrations and whose weak equivalences are the intersection of weak equivalences in $M_1$ and $M_2$. This model structure is a right Bousfield delocalization of both $M_1$ and $M_2$.