We construct two non-isomorphic nuclear,
stably finite, real rank zero C*-algebras
E and E'
for which there is an isomorphism of ordered groups
that is compatible with all the coefficient transformations.
The C*-algebras E and E' are not isomorphic since
there is no map as above that is also compatible
with the Bockstein operations.
By tensoring with Cuntz's algebra Oinfinity one obtains a pair of
non-isomorphic, real rank zero, purely infinite C*-algebras
with similar properties.