  We introduce a spinful variant of the Sachdev-Ye-Kitaev model with an
effective time reversal symmetry, which can be solved exactly in the limit of a
large number $N$ of degrees of freedom. At low temperature, its phase diagram
includes a compressible non-Fermi liquid and a strongly-correlated spin singlet
superconductor that shows a tunable enhancement of the gap ratio predicted by
BCS theory. These two phases are separated by a first-order transition, in the
vicinity of which a gapless superconducting phase, characterized by a non-zero
magnetization, is stabilized upon applying a Zeeman field. We study equilibrium
transport properties of such superconductors using a lattice construction, and
propose a physical platform based on topological insulator flakes where they
may arise from repulsive electronic interactions.
