Just noticed this on my store bought plant (not grown from seed). It's now fruiting, and I don't see any white spots on the leaves.
Update: 25 Feb 2016 with close up of the stems and an more distant view of the plant health
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That appears to be Southern Blight, Sclerotium rolfsii
It's a fungal disease from the soil, and affects many species of plants. You cannot cure plants infected with it. From the University of California:
- Rotate to nonhost crops, such as corn, sorghum, rice, or small grains, for at least 2 years to reduce inoculum.
- Deep plowing to bury plant refuse may help to destroy sclerotia.
- Keeping the tops of beds dry in tomato fields helps reduce the disease in furrow- and buried-drip-irrigated fields.
For potted plants, I'd recommend disposing of the potting medium, and sterilizing or disposing of the container.