You can apparently produce a graft of one onto the other, and there's apparently another technique to create a combined plant.
https://en.wikipedia.org/wiki/Pomato
The pomato (a portmanteau of potato and tomato), also known as a tomtato, is a hybrid plant that is able to grow both tomatoes and potatoes. The most common method of creating a pomato is grafting together a tomato plant and a potato plant, both of which are members of the Solanum genus in the Solanaceae (nightshade) family. Another method is somatically fusing the two plants together.[1] Cherry tomatoes grow on the vine, while white potatoes grow in the soil from the same plant.[2]
The concept of grafting related potatoes and tomatoes so that both are produced on the same plant dates back to at least 1833.[3]
As with all grafts, this plant will not occur in nature and cannot be grown from seed, because the two parts of the plant remain genetically separate, and only rely on each other for nourishment and growth.
The somatic fusion of potato and tomato cells is also possible, though this plant cannot produce fertile seeds. The first such somatic hybrid was bred in 1978.[1][7]
I've never heard of somatic fusion before now.
Grafted pomato plants were launched in the United Kingdom in September 2013 by the horticultural mail-order company Thompson & Morgan, who sold pre-grafted plants branded as the "TomTato". The Incredible Edible nursery in New Zealand announced a "DoubleUP Potato Tom" in the same month.
