It is well known that strong concurrent precipitation, mainly located along grain/subgrain boundaries, results in a coarse and elongated grain structure, and unusual sharp P {011} <566> and/or M {113} <110> texture components during annealing of supersaturated AA3xxx-series alloys. In this study, a supersaturated as-cast Al–Mn–Fe–Si alloy was annealed at three temperature–time paths after cold rolling and their effects on the softening behavior have been analyzed and compared in terms of final grain structure and texture. In particular, material conditions with dispersoids located along grain/sub-grain boundaries were produced prior to annealing. The effects of pre-existing dispersoids as well as dispersoids formed during annealing (concurrent precipitation), both of which are heterogeneously distributed along grain/sub-grain boundaries, on the recrystallization behavior of the deformed material were investigated and compared. The results clearly show that, due to their larger sizes, these pre-existing dispersoids do not significantly increase the strength of the P/M texture components as compared to the dispersoids formed during annealing, even though the former can still affect the grain structure. It can be concluded that sharp P/M textures develop when recrystallization takes place at low temperature where nucleation of other orientations, whether from particle stimulated nucleation (PSN) or other nucleation mechanisms are suppressed by concurrent precipitation.