  We establish a rationality result for linear combinations of traces of cycle
integrals of certain meromorphic Hilbert modular forms. These are meromorphic
counterparts to the Hilbert cusp forms $\omega_m(z_1,z_2)$, which Zagier
investigated in the context of the Doi-Naganuma lift. We give an explicit
formula for these cycle integrals, expressed in terms of the Fourier
coefficients of harmonic Maass forms. A key element in our proof is the
explicit construction of locally harmonic Hilbert-Maass forms on
$\mathbb{H}^2$, which are analogous to the elliptic locally harmonic Maass
forms examined by Bringmann, Kane, and Kohnen. Additionally, we introduce a
regularized theta lift that maps elliptic harmonic Maass forms to locally
harmonic Hilbert-Maass forms and is closely related to the Doi-Naganuma lift.
