  We extend the recently proposed $SU(5) \times \mathcal{T}_{13}$ model for the
asymmetric texture to the up-type quark and seesaw sectors. The hierarchical
up-type quark masses are generated from higher-dimensional operators involving
family-singlet Higgses, gauge-singlet familons, and vectorlike messengers. The
complex-tribimaximal (TBM) seesaw mixing arises from the vacuum structure of a
minimal number of familons, resulting in an alignment between the Yukawa and
Majorana matrices of the seesaw formula. Introducing four right-handed
neutrinos, normal ordering of the light neutrino masses is obtained, with
$m_{\nu_1} = 27.6\ \mathrm{meV}$, $m_{\nu_2} = 28.9\ \mathrm{meV}$ and
$m_{\nu_3} = 57.8\ \mathrm{meV}$. Their sum almost saturates Planck's
cosmological upper bound ($120$ $\text{meV}$). The right-handed neutrino masses
are expressed in terms of two parameters for a particular choice of familon
vacuum alignment. We predict the $\require{cancel}\cancel{CP}$
Jarlskog-Greenberg invariant to be $|\mathcal{J}| = 0.028$, consistent with the
current PDG estimate, and Majorana invariants $|\mathcal{I}_1| = 0.106$ and
$|\mathcal{I}_2| = 0.011$. A sign ambiguity in the model parameters leads to
two possibilities for the invariant mass parameter $|m_{\beta \beta}|$: $13.02$
or $25.21$ $\text{meV}$, both within an order of magnitude of the most rigorous
experimental upper limit ($61$--$165$ $\text{meV}$).
