  The purpose of this paper is to solve a kind of Riemann-Hilbert boundary
value problem for $(\varphi,\psi)$-harmonic functions, which are linked with
the use of two orthogonal basis of the Euclidean space $\mathbb{R}^m$. We
approach this problem using the language of Clifford analysis for obtaining the
explicit expression of the solution of the problem in a Jordan domain
$\Omega\subset\mathbb{R}^m$ with fractal boundary. One of the remarkable
feature in this study is that the boundary data involves higher order Lipschitz
class of functions.
