The MV Liemba connects Kigoma with Mpulungu in Zambia every second Wednesday (1st/2nd/economy class US$105/95/75, at least 40 hours). US dollars cash only.
En route, it stops at various lakeshore villages, including Lagosa ($40/35/30, for Mahale Mountains National Park), Kalema ($55/45/40, southwest of Mpanda), Kipili ($75/65/55) and Kasanga ($100/90/70, southwest of Sumbawanga). In theory, departures from Kigoma are on Wednesday at 4pm, reaching Mpulungu Friday morning. Departures from Mpulungu are (again, in theory) on Friday afternoon at about 2pm, arriving back in Kigoma on Sunday afternoon, but these often change. The boat was undergoing an overhaul in 2017, with interruptions and changes to the schedule.
Be aware that your luggage may be searched carefully before or during boarding for contraband or souvenirs purchased without receipts showing paid VAT.