The harmonicity of a smooth map from a Riemann surface into the $6$-dimensional sphere $S^6$ amounts to the closeness of a certain $1$-form that can be written in terms of the nearly Kähler structure of $S^6$. We will prove that the immersions $F$ in $\mathbb{R}^7$ obtained from superconformal harmonic maps in $S^3\subset S^6$ by integration of the corresponding closed $1$-forms are isothermic. The isothermic surfaces so obtained include a certain class of constant mean curvature surfaces in $\mathbb{R}^3$ that can be deformed isometrically through isothermic surfaces into non-spherical pseudo-umbilical surfaces in $\mathbb{R}^7$.