Faithful interpretations of Matveev's virtual manifolds

S. V. Matveev introduced the concept of virtual manifolds and posed the question of whether the natural map from the set of virtual manifolds to the set of $3$-manifolds with $RP^2$-singularities is injective. We prove that this map is not injective, but its refining map from the set of virtual manifolds to the set of $3$-manifolds with $RP^2$-singularities equipped with collections of arcs of a special type is injective.