The bacteria that causes Tuberculosis is a nasty little beast. The white blood cells that clear infection in your body work by ingesting bacteria and then breaking them up, and the TB escapes this by letting itself get ingested and then sitting inside your white blood cells. They don’t sit passively, however, they burst out of the cell and recruit a whole host of other blood cells which surround the infection and form what’s called a granuloma. The bacteria stay inside the granuloma and become dormant, but if they escape they can set up other sites of infection throughout the body.
The white blood cells that first ingest the TB bacteria are macrophages, which kill invading particles (be they cells or bits of cells, or even dead parts of your own cells) by ingesting them into a vacuole and then breaking them down. The TB gets ingested fine, but once inside the cell, it stops the cell from breaking it down. That means that you now have a white blood cells infected with a TB bacterium.