Electric field due to a dipole sheet

Question

The potential due to a dipole sheet, $S$, having dipole moment density, $\vec\tau$ (per unit area), is given by $$ \phi(\mathbf r) = \frac{1}{4\pi\epsilon_0}\int_S\frac{\vec\tau(\mathbf r')\cdot(\mathbf{r-r'})}{|\mathbf{r-r'}|^3} dS'. $$

Now, if I naïvely calculate $\mathbf E$ as $-\nabla\phi$ (as $\nabla$ operates only on $\mathbf r$, it will pass through the integral), then I'll get Dirac's delta in the integrand, meaning that $\mathbf E$ is $0$ everywhere ($\mathbf r$ not on the sheet, $S$), which is obviously wrong.

Question: What has gone awry? The same method yields correct results for electric field due to a charged sheet.

Answer 1

In analogy with dipole charges, a dipole sheet can be approximated as two infinite parallel sheets of opposite charge density separated by some distance. What is the electric field in the volume not between the two sheets?

Now let the sheets come together while keeping the dipole moment per unit area constant. Does anything change about the electric field?