Driving on the “other” side of the road was the least their problems. One wayward driver in Mexico was certain his wrong turn had invited a police shakedown. In Turkey, one breakdown among many ended with locals helping build a campfire on a frigid night. In Kenya, a repair took on extra urgency when the travelers realized they were in the middle of a path for drug smugglers.

Getting behind the wheel in a foreign country can give tourists greater freedom to explore, but also more ways for things to go awry. For many travelers, the extra adventures are worth the troubles.

It was a torrid, humid day in La Paz, Mexico, two years ago when Chris Collard accidentally turned his truck the wrong way down a one-way road. He quickly realized his mistake and reversed out, but a police officer noticed his error.

“I thought, here we go,” said Mr. Collard, whose experiences with the police in other countries have not always been pleasant. An international photojournalist and owner of Adventure Architects, he was expecting to be forced to pay a bribe.