Mix cream cheese, mozzarella, and Parmesan in a stand mixer using the paddle attachment on low speed until softened.
Squeeze all of the water out of the thawed spinach, and add it to the mixing bowl along with mayonnaise, tomatoes, artichokes, garlic, and spices. Continue to mix.
Scrape down the bowl, add the heavy cream, and continue to mix until combined.
To heat, spread the dip in an oven-safe dish and bake for 7-10 minutes. The top should start to darken, and the dip should be bubbling around the edges.