How does a cloud of dry ash throw lightning with no rainstorm?

The rock does the work. In an explosive eruption molten rock shatters into ash at speed, and the new surfaces carry unbalanced charge while particles collide millions of times and transfer electrons. Then the winds sort particles by size into regions of opposite charge, and when the electric field grows strong enough the air breaks down. In tall columns ice forms too, and a second mechanism joins in.

Lightning in a cloud of dry ash looks like a contradiction, because in an ordinary storm the charging rests on droplets and on ice. Here the work is done in rock.

In an explosive eruption molten rock shatters into ash at speed, and the new surfaces carry unbalanced charge. Particles collide millions of times and transfer electrons, like a balloon charged on hair, in an enormous volume.

Scattered charge is still not lightning. The winds sort particles by size into different regions, each group tending toward a different charge, and when the field grows strong enough the air breaks down.

In tall ash columns a second mechanism joins in: ice and small hail rub together and move charge as in a thunderstorm. So picking up lightning can give distant warning of dangerous ash even when cloud hides the eruption. Several charging mechanisms operate together, and some change with grain size, with humidity and with the height of the column.