  We propose a class of topological superconductivity in which the pairing
order is $\mathbb{Z}_2$ topologically obstructed in a three-dimensional
time-reversal invariant system. When two Fermi surfaces are related by
time-reversal and mirror symmetries, such as those in a $\mathbb{Z}_2$ Dirac
semimetal, the inter-Fermi-surface pairing in the weak-coupling regime inherits
the band topological obstruction. As a result, the pairing order cannot be
well-defined over the entire Fermi surface and forms a time-reversal invariant
generalization of U($1$) monopole harmonic pairing. A tight-binding model of
the $\mathbb{Z}_2$ topologically obstructed superconductor is constructed based
on a doped $\mathbb{Z}_2$ Dirac semimetal and exhibits nodal pairings. At an
open boundary, the system exhibits a time-reversal pair of topologically
protected surface states.
