  We introduce a family of tensor quantum-mechanical models based on
irreducible rank-$3$ representations of $\mathrm{Sp}(N)$. In contrast to
irreducible tensor models with $\mathrm{O}(N)$ symmetry, the fermionic
tetrahedral interaction does not vanish and can therefore support a melonic
large $N$ limit. The strongly-coupled regime has a very analogous structure as
in the complex SYK model or in
$\mathrm{U}(N)\times\mathrm{O}(N)\times\mathrm{U}(N)$ tensor quantum mechanics,
the main difference being that the states are now singlets under
$\mathrm{Sp}(N)$. We introduce character formulas that enumerate such singlets
as a function of $N$, and compute their first values. We conclude with an
explicit numerical diagonalization of the Hamiltonian in two simple examples:
the symmetric model at $N=1$, and the antisymmetric traceless model at $N=3$.
