What does one have to do to make orange juice that isn’t Kosher?
Peel the oranges with a muslim scimitar?
Scrub the bottles with hog hair brushes or use soap made from lard would do it. The Rabbi must bless the food selection and the method of processing the food.
I think many manufacturers simply add labels as assurance that the food is acceptable during Passover. However, one can never be sure with a religion that has to follow over 600 rules to please God.
Add animal (pork sourced) gelatin as a thickener.
Process on equipment that is shared with processing treif products (gravy from meats that weren't properly slaughtered and contain blood, or a shellfish juice product like Clamato, for two examples)
Processing on equipment lubricated with animal fats (read lard).