You have some form of fungal disease--It may be blight.

I do know that the first line of defense is to remove all the infected leaves. You then need a fungicide and I can't advise you there either. When watering try not to get the leaves wet.

Most leaf diseases spread the same way. Water splashes on the infected leaf, which splashes on to an uninfected leaf. The water carries the spores of the disease to the new leaf and it becomes infected. Eventually the whole plant dies. In addition some diseases live in the soil.

In that case you have to solarize the soil by putting black plastic on it and let the sun kill it.

Sorry to be the bearer of bad news, but maybe you can get rid of it if you are vigilant

Thanks for the info. Even if it's bad news, I'm happy to get the input.

I'll remove the affected leaves today (though the plants may not survive, as the disease seems present on most leaves) and look into the other treatment options.