Harmonic functions with highly intersecting zero sets

We show that the number of isolated zeros of a harmonic map $h:{\mathbb{R}}^2\to {\mathbb{R}}^2$ inside the ball of radius $r$ can grow arbitrarily fast with $r$, while its maximal modulus grows in a controlled manner. This result is an analogue, in the context of harmonic maps, of the celebrated Cornalba-Shiffman counterexamples to the transcendental Bézout problem.