For Mandie Adams, the complete solar eclipse on Dec. 4 will be the twelfth one particular she’s witnessed. It will also be the twelfth for her teddy bear, Dinky Doo.

Viewing it will not be straightforward. The eclipse will be seen only on a sliver of Earth in Antarctica, the South Orkney Islands and the surrounding ocean. Ms. Adams, a rental-property operator who lives in Southend-on-Sea, England, flew from London to Madrid to Buenos Aires to the city of Ushuaia on the southern tip of Argentina. From there, she will board a fifteen-day eclipse cruise, which sails as a result of the notoriously turbulent Drake’s Passage, to see just underneath two minutes of complete eclipse darkness—if there are not clouds.