Recent experiments in multiband Fe-based and heavy-fermion superconductors have challenged the long-held dichotomy between simple $s$- and $d$-wave spin-singlet pairing states. Here, we advance several time-reversal-invariant irreducible pairings that go beyond the standard singlet functions through a matrix structure in the band/orbital space, and elucidate their naturalness in multiband systems. We consider the $stau_{3}$ multiorbital superconducting state for Fe-chalcogenide superconductors. This state, corresponding to a $d+d$ intra- and inter-band pairing, is shown to contrast with the more familiar $d +text{i}d$ state in a way analogous to how the B- triplet pairing phase of enhe superfluid differs from its A- phase counterpart. In addition, we construct an analogue of the $stau_{3}$ pairing for the heavy-fermion superconductor CeCu$_{2}$Si$_{2}$, using degrees-of-freedom that incorporate spin-orbit coupling. Our results lead to the proposition that $d$-wave superconductors in correlated multiband systems will generically have a fully-gapped Fermi surface when they are examined at sufficiently low energies.
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