  We calculate the prominent perturbative contributions shaping the one-loop
scalar spectrum of the minimal non-supersymmetric renormalizable $SO(10)$ Higgs
model whose unified gauge symmetry is spontaneously broken by an adjoint
scalar. Focusing on its potentially realistic $45\oplus 126$ variant in which
the rank is reduced by a VEV of the 5-index self-dual antisymmetric tensor, we
provide a thorough analysis of the corresponding one-loop Coleman-Weinberg
potential, paying particular attention to the masses of the potentially
tachyonic pseudo-Goldstone bosons (PGBs) transforming as $(8,1,0)$ and
$(1,3,0)$ under the Standard Model gauge group. The results confirm the assumed
existence of extended regions in the parameter space supporting a locally
stable SM-like quantum vacuum inaccessible at the tree-level. The effective
potential (EP) tedium is compared to that encountered in the previously studied
$45\oplus 16$ $SO(10)$ Higgs model where the polynomial corrections to the
relevant pseudo-Goldstone masses turn out to be easily calculable within a very
simplified purely diagrammatic approach.
