Not quite sure of what you are asking. Is this it?

There is rather famous example: u(x,y)=x^2 – y^2 and v(x,y)=2xy.

Both are harmonic, because they are the standard u+iv decomposition of z^2.

Moreover, v is the harmonic conjugate of u. BUT u is not the harmonic conjugate of v, because 2xy+i(x^2 - y^2) is nowhere analytic.