  Dimer models have long been a fruitful playground for understanding
topological physics. Here we introduce a new class - termed Majorana-dimer
models - wherein bosonic dimers are decorated with pairs of Majorana modes. We
find that the simplest examples of such systems realize an intriguing,
intrinsically fermionic phase of matter that can be viewed as the product of a
chiral Ising theory, which hosts deconfined non-Abelian quasiparticles, and a
topological $p_x - ip_y$ superconductor. While the bulk anyons are described by
a single copy of the Ising theory, the edge remains fully gapped. Consequently,
this phase can arise in exactly solvable, frustration-free models. We describe
two parent Hamiltonians: one generalizes the well-known dimer model on the
triangular lattice, while the other is most naturally understood as a model of
decorated fluctuating loops on a honeycomb lattice. Using modular
transformations, we show that the ground-state manifold of the latter model
unambiguously exhibits all properties of the $\text{Ising} \times (p_x-ip_y)$
theory. We also discuss generalizations with more than one Majorana mode per
site, which realize phases related to Kitaev's 16-fold way in a similar
fashion.
