In late 2017, with prices of cryptocurrencies like bitcoin surging, Coinbase CEO Brian Armstrong faced a tough choice: Would his company, the leading U.S. cryptocurrency exchange, become more like a traditional financial services company or would it lean into the industry’s wild side?

In the end, Mr. Armstrong hired deputies who pursued both paths, a decision that divided the company’s leadership into competing camps. The clash came to a head in May of this year, when the two deputies—President and Chief Operating Officer Asiff Hirji and Chief Technology Officer Balaji Srinivasan—left the company within a month of each other.

At times, the conflict between the two men, which hasn’t previously been reported, became so intense that their interactions devolved into shouting matches, according to two people who observed the episodes and others who were told about them. Mr. Hirji, a former executive from stock brokerage TD Ameritrade who was accustomed to working in large organizations, lobbied for Coinbase to prioritize attracting financial institutions to its services. Meanwhile, Mr. Srinivasan, an experienced Silicon Valley startup founder, wanted Coinbase to cater more to individuals by aggressively expanding the types of digital currencies in its exchange, including some that occupy a regulatory gray area.