Filtering an array with a deeply nested array in JS

How can I filter an array with a deeply nested array? Given the following 2 arrays, I need the result to be an array with only the rice cakes and the gluten-free-pizza objects:

const foodsILike = ['gluten-free', 'carb-free', 'flavor-free'];
const foodsAvailable = [
  { name: 'pasta', tags: ['delicious', 'has carbs']}, 
  { name: 'gluten-free-pizza', tags: ['gluten-free']}, 
  { name: 'pizza', tags: ['delicious', 'best meal of the year']},
  { name: 'rice cakes', tags: ['flavor-free']}
]

I tried the following which just returns everything (the 4 objects):

var result = foodsAvailable.filter(function(food) {
  return foodsILike.filter(function(foodILike) {
    return foodILike === food;
  })
})

result
// Array [ Object, Object, Object, Object]