A functorial classification, up to approximate unitary equivalence, is given of unital real *-homomorphisms from a real C*-algebra arising as an inductive limit of real forms on finite direct sums of matrix algebras over the continuous complex valued functions on the unit interval to another such algebra. The invariant consists of a diagram Cu(A)-> Cu(A ⊗_R C)-> Cu(A ⊗_R H) of Cuntz semigroups with distinguished elements. As a corollary, a classification, up to *-isomorphism, of such real approximate interval algebras is obtained. Also, unital real *-homomorphisms from real approximately finite dimensional C*-algebras to a certain general class of real C*-algebras are classified, up to approximate unitary equivalence, by a diagram K₀(A)-> K₀(A ⊗_RC)-> K₀(A ⊗_RH) of ordered K₀ groups with distinguished elements. As a corollary, a new proof of the already known classification of real approximately finite dimensional algebras in terms of this invariant is obtained.

Research funded by The Natural Sciences and Engineering Research Council of Canada. The first named author would like to thank the Westfälische Wilhelms-Universität Münster for their hospitality during his sabbatical year, when much of this work was completed.