  Determining the exact pairing symmetry of the superconducting order parameter
in candidate unconventional superconductors remains an important challenge.
Recently a new method, based on phase sensitive quasiparticle interference
measurements, was developed to identify gap sign changes in isotropic multiband
systems. Here we extend this approach to the single-band and multiband nodal
$d$-wave superconducting cases relevant, respectively, for the cuprates and
likely for the infinite-layer nickelate superconductors. Combining analytical
and numerical calculations, we show that the antisymmetrized correction to the
tunneling density of states due to nonmagnetic impurities in the Born and
intermediate-scattering limits shows characteristic features for sign-changing
and sign-preserving scattering wavevectors, as well as for the
momentum-integrated quantity. Furthermore, using a realistic approach
accounting for the Wannier orbitals, we model scanning tunneling microscopy
data of $\text{Bi}_2\text{Sr}_2\text{CaCu}_2\text{O}_{8+\delta}$, which should
allow the comparison of our theory with experimental data.
