As noted in the Out of Supply section of Chapter 12: "If broken by damage when out of supply, an Explorer dies the next time another supply point should be spent."
As I read this, any explorer that is currently in the state of broken by damage dies the next time the group needs to spend supply. However in my latest session (during which the crew ran out of supply while trekking back to base camp), the Crew would merely perform first aid as indicated in Chapter 4, page 66 any time an explorer became broken by damage. In this way they were effectively able to stave off death repeatedly despite being entirely out of supply.
I'm wondering if this is an intended interaction. Can you do first aid on someone broken by damage due to supply?