Soft magnetorheological elastomers (MREs) are multifunctional composites consisting of magnetic particles embedded in an elastomeric matrix (stiffness below 10 kPa). When small magnetic fields are applied, significant magneto-mechanical coupling between the phases can be achieved. In addition, the use of hard-magnetic particles allows the composite to be permanently magnetized and establish a magnetic field even in the absence of external magnetic actuation. Some state-of-the-art applications of soft polymers (e.g., skin patches, bio-adhesive skin sensors, and meshes for wound closure undergoing finite deformations) foster the study of their failure mechanisms. In this regard, a point of interest relates to the strategies to enhance the fracture resistance and to arrest the propagation of cracks. On these basis, we examine how remanent magnetic fields impact the fracture behavior of ultra-soft MREs with pre-existing cuts. We show that MREs with remanent magnetization absorb up to a 50% more fracture energy than non-pre-magnetized samples. Digital image correlation allows to calculate the full strain fields at the crack tip and estimate the J-integral. Moreover, we report crack closure due to the magnetic field as a phenomenon that delays the opening of cracks. To further understand these findings, a bespoke non-local phase-field model incorporating magneto-mechanical coupling is developed. The results indicate that remanent magnetic fields induce beneficial compressive stresses in the medium which decrease the stress concentration at the crack tip. Overall, we unveil intriguing potential applications for functional actuators with enhanced fracture performance and self-healing capacities.