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For each $k = 0,\dots,n$ we construct a continuous phase $f_k$, with $f_k(0)= (n-2k)\frac{\pi}{2}$, and viscosity sub- and supersolutions $v_k$, $u_k$, ofthe elliptic PDE $\sum_{i=1}^n \arctan(\lambda_i(D^2 w)) = f_k(x)$ such that$v_k-u_k$ has an isolated maximum at the origin.  It has been an open question whether the comparison principle would hold inthis second order equation for arbitrary continuous phases $f\colon\Omega\to(-n\pi/2,n\pi/2)$. Our examples show it does not.

Counterexamples to the comparison principle in the special Lagrangian potential equation