It is widely accepted that the Everett's (or "many-worlds") interpretation of quantum mechanics is the only one which is appropriate for quantum cosmology because no environment may exist for Universe as a whole. We discuss, in the framework of the Everett's interpretation, the (quasi-) classical stage of evolution of the Universe when there coexist "classically incompatible" configurations of matter, or classical alternative realities ("alternatives" for short). In the framework of the Everett's interpretation the semiclassical gravity (where the gravitational field is classical and the non-gravitational fields are quantum) is more natural than theories including quantizing gravitational field. It is shown that the semiclassical (at least on the astrophysical and cosmological scales) Everett-type gravity leads to the observational effect known as the effect of dark matter. Instead of assuming special forms of matter (weakly interacting with the known matter), the role of the dark matter is played in this case by the matter of the usual kind which however belongs to those alternative realities (Everett's worlds) which remain «invisible», i.e. not perceived with the help of non-gravitational fields.