  Populations of heterogeneous phase oscillators with frustrated random
interactions exhibit a quasi-glassy state in which the distribution of local
fields is volcano-shaped. In a recent work [Phys. Rev. Lett. 120, 264102
(2018)] the volcano transition was replicated in a solvable model using a
low-rank, random coupling matrix $\mathbf M$. We extend here that model
including tunable nonreciprocal interactions, i.e. ${\mathbf M}^T\ne \mathbf
M$. More specifically, we formulate two different solvable models. In both of
them the volcano transition persists if matrix elements $M_{jk}$ and $M_{kj}$
are enough correlated. Our numerical simulations fully confirm the analytical
results. To put our work in a wider context, we also investigate numerically
the volcano transition in the analogous model with a full-rank random coupling
matrix.
