A word anticorrespondence problem
Isn't the solution a one-counter language? 

The succession of the domino's is easily seen as a finite state restriction. For the unequality of $u'$ and $v'$ you have to find a single position where these strings differ. Keep track of $|u'|-|v'|$ during computation on the stack. You just nondeterministically choose such a letter, on a position of the longest of $u'$ and $v'$. Then keep simulating until the shortest string comes alongside with that same position.

I know this construction from a short paper (Prefix and equality languages of rational functions are co-context-free IPL 28 (1988) 77-79): "Theorem. For every two rational functions F and G, the complement of their prefix language Pref(F,G) is a one-counter language". 
However that technique is a slight variation from an earlier result by J. Berstel: Every iterated morphism yields a co-cfl, Inform. Process. Lett. 22 (1986) 7-9.  

Provided of course, I read your definition in the right way.