Phallus impudicus really are nothing more than a type of stinkhorn mushroom that appears during cool, wet weather, mainly in fall and winter. They start with a white "egg" that is partially visible above soil, with a network of roots anchoring it below. Once the mushroom is mature, the full-grown stalk emerges fast — sometimes within an hour.
The vile smell of a mature Stinkhorn might be taken to suggest that these fungi are toxic or at least inedible; however, some people do eat them, but only at the 'egg' stage when the odour is not so evident.