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We investigate the Einstein vacuum equations as well as the Einstein-null fluid equations describing neutrino radiation. We find new structures in gravitational waves and memory for asymptotically-flat spacetimes of slow decay. These structures do not arise in spacetimes resulting from data that is stationary outside a compact set. Rather the more general situations exhibit richer geometric-analytic interactions displaying the physics of these more general systems. It has been known that for stronger decay of the data gravitational wave memory is finite and of electric parity only. We investigate general spacetimes that are asymptotically flat in a rough sense, where the decay of the data to Minkowski space towards infinity is very slow. Main new feature: We prove that there exists diverging magnetic memory sourced by the magnetic part of the curvature tensor (a) in the Einstein vacuum and (b) in the Einstein-null-fluid equations. The magnetic memory occurs naturally in the Einstein vacuum setting (a) of pure gravity. In case (b), in the ultimate class of solutions, the magnetic memory also contains a curl term from the energy-momentum tensor for neutrinos also diverging at the highest rate. The electric memory diverges too, it is generated by the electric part of the curvature tensor and in the Einstein-null-fluid situation also by the corresponding energy-momentum component. In addition, we find a panorama of finer structures in these manifolds. Some of these manifest themselves as additional contributions to both electric and magnetic memory. Our theorems hold for any type of matter or energy coupled to the Einstein equations as long as the data decays slowly towards infinity and other conditions are satisfied. The new results have many applications ranging from mathematical general relativity to gravitational wave astrophysics, detecting dark matter and other topics in physics.