  We show that every ring isomorphism between the algebras of locally
measurable operators for type II$_\infty$ von Neumann algebras is similar to a
real $^*$-isomorphism. This together with previous results by the author and
Ayupov--Kudaybergenov completely describes ring isomorphisms between the
algebras of locally measurable operators as well as lattice isomorphisms
between the projection lattices for a general pair of von Neumann algebras
without finite type I direct summands.
