TEL AVIV—Israel’s first large gas field, Tamar, is due to begin producing natural gas next April. It is an economic bonanza for the state, and a security nightmare for the navy, tasked with protecting the huge area, much of which is outside Israel’s territorial waters.

“These fields have strategic significance and could be easily a target for our neighbors,” a senior naval official in charge of planning, told The Media Line in an exclusive briefing in his office in Tel Aviv. “Usually to protect an area, we just make a sterile zone around it. But we can’t do that in internation...