The intrinsic spins and their correlations are the least understood characteristics of fission fragments from both theoretical and experimental points of view. In many nuclear reactions, the emerging fragments are typically excited and acquire an intrinsic excitation energy and an intrinsic spin depending on the type of the reactions and interaction mechanism. Both the intrinsic excitation energies and the fragments' intrinsic spins and parities are controlled by the interaction mechanism and conservations laws, which lead to their correlations and determines the character of their deexcitation mechanism. We outline here a framework for the theoretical extraction of the intrinsic spin distributions of the fragments and their correlations within the fully microscopic real-time density-functional theory formalism and illustrate it on the example of induced fission of ^{236}U and ^{240}Pu, using two nuclear energy density functionals. These fission fragment intrinsic spin distributions display new qualitative features previously not discussed in literature. Within this fully microscopic framework, we extract for the first time the intrinsic spin distributions of fission fragments of ^{236}U and ^{240}Pu as well as the correlations of their intrinsic spins, which have been debated in literature for more than six decades with no definite conclusions so far.