A potential reply to avoid coming across poorly could be something like: "what time code or project number should I report my time under when I stop my vacation and go on-call?"

This is somewhat cultural: it would not be considered unacceptable in many startups in the US, where I am right now.

But your username and grammar make me wonder whether you're French and in France, are you? If you are, I think it may actually be an illegal request. European attitudes towards work are extremely different to US ones, which are themselves more relaxed than e.g. East Asian.

From https://newatlas.com/right-to-disconnect-after-hours-work-em... :

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"France, in particular, has been ahead of the world in establishing legal frameworks protecting a person's right to disconnect. Back in 2001 the idea was first floated when the French Supreme Court ruled that employees are under no obligation to bring work home, and as technology progressed the Court continued to update its ruling. In 2004, for example, it was established that it was not misconduct if an employee was not reachable on a smartphone outside of work hours.

The right to disconnect was solidified at the beginning of 2017 with France introducing the El Khomri law, which suggests every employee contract must include a negotiation of obligations required of an employee regarding how connected they are outside of office hours. The law is reasonably vague and doesn't restrict after-hours work communication, but rather obliges organizations to negotiate these terms clearly with prospective employees."

If you have to work during your holidays then they shouldn't be called holidays.