I doubt it will be compulsory. But I suspect that that health passports will be a de facto requirement enforced by employers, private companies and organisations because their staff, customers and shareholders demand it. For example, to fly, land in another country, go to a supermarket, the cinema, university, place of employment etc.
The new narrative amongst anti-vaxxers will be that these things have not been adequately tested - missing the point that the intensity and focus on testing has effectively accelerated that process by concentrating additional resources and man-hours. Because