Watermixable Oil Paint
They are not water-based, but water-soluble paint and that distinction is key. These paints are real oils, they simply have the ability to be mixed and cleaned with water.
Water soluble means that you can use water to thin the oil paint (though traditional oil mediums like linseed or stand oils can also be used). While we learned in elementary school that water and oil do not mix, the water mixable (also called water miscible or solvent-free oils) paints have been formulated to accept water and get around that chemical barrier.
These paints retain almost all of the aspects which painters love about oil paints. They make it more accessible, convenient, and easier to work with oils. With practice and attention to how they are mixed, water mixable oils can produce stunning paintings that rival the color depth and texture of those done with traditional oils.