  Let $\mathrm{pod}_{-4}(n)$ denote the number of partition quadruples of $n$
where the odd parts in each partition are distinct. We find many arithmetic
properties of $\mathrm{pod}_{-4}(n)$ involving the following infinite family of
congruences: for any integers $\alpha \ge 1$ and $n \ge 0$,
  \[\mathrm{pod}_{-4}\Big({{3}^{\alpha +1}}n+\frac{5\cdot {{3}^{\alpha
}}+1}{2}\Big)\equiv 0 \pmod{9}.\] We also establish some internal congruences
and some congruences modulo 2, 5 and 8 satisfied by $\mathrm{pod}_{-4}(n)$.
