Tomatoes dying from the bottom up
I recently moved to Malawi, Africa, started a garden, and promptly started cherry tomatoes b/c they are very difficult to find in the market. They grew nicely for the first couple weeks, but then as the main stem continued to grow and produce new leaves and tomato bunches, the stems and leaves on the base of the plant started browning and, ultimately, dying. It's been a slow chase ever since: The plants grow upward but the death is chasing it from below.
I haven't been able to find any pests on the ground or on the plants. Other plants are growing successfully in the same soil. Any ideas?