Type I superstring theory is unoriented, and it seems that it needs to be so in order to exist.
Now, we always have open-closed duality, that connects at least the ultraviolet sector of a theory with the infrared sector of another, so in principle we should have some oriented open strings coming from it, by duality with a closed oriented theory.
And we have all the D-brane stuff, that surely produces another kinds of open string theories with the restriction of terminating there in the branes.
¿Is some of these D-brane examples an oriented theory? ¿Is there some canonical, well known, example of a consistent oriented theory with open strings?