  We consider the following question: Let $S_1$ and $S_2$ be two smooth,
totally-real surfaces in $\mathbb{C}^2$ that contain the origin. If the union
of their tangent planes is locally polynomially convex at the origin, then is
$S_1 \cup S_2$ locally polynomially convex at the origin? If $T_0S_1 \cap
T_0S_2=\{0\}$, then it is a folk result that the answer is yes. We discuss an
obstruction to the presumed proof, and provide a different approach. When
dimension of $T_0S_1 \cap T_0S_2$ over the field of real numbers is 1, we
present a geometric condition under which no consistent answer to the above
question exists. We then discuss conditions under which we can expect local
polynomial convexity.
