String complexity is defined as the cardinality of a set of all distinct words (factors) of a given string. For two strings, we define joint string complexity as the set of words that are common to both strings. We also relax this definition and introduce joint semi-complexity restricted to the common words appearing at least twice in both strings. String complexity finds a number of applications from capturing the richness of a language to finding similarities between two genome sequences. In this paper we analyze joint complexity and joint semi-complexity when both strings are generated by a Markov source. The problem turns out to be quite challenging requiring subtle singularity analysis and saddle point method over infinity many saddle points leading to novel oscillatory phenomena with single and double periodicities.