Post Undeleted by Jack Huizenga
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EDIT: New example, hopefully this one works.

Blow up $\mathbb{P}^2$ at a point $p$, then blow up the resulting surface at a point $q$ on the exceptional divisor. The resulting surface has Picard group generated by the class $H$ of a line, the proper transform $E_1$ of the first exceptional divisor, and the second exceptional divisor $E_2$. We have $E_1^2 = -2$ and $E_2^2 = -1$, while $E_1\cdot E_2 = 1$. Now consider $D = E_1 +2E_2$. The Zariski decomposition of $D$ is $P = 0$, $N = E_1+2E_2$, as no subdivisor effective divisor supported on $E_1$ and $E_2$ is nef. Then $D\cdot E_1 = 0$. We can get strict inequality instead by taking $D = E_1+3E_2$, for instance.

2 added 175 characters in body

Let $X\to \mathbb{P}^2$ be the blowup of

Blow up $\mathbb{P}^2$ at one a point $p$. Let $D$ be p$, then blow up the full preimage of resulting surface at a line in$\mathbb{P}^2$passing through the point$p$, consisting of q$ on the exceptional divisor. The resulting surface has Picard group generated by the class $E$ and H$of a line, the proper transform$H-E$E_1$ of the line through first exceptional divisor, and the second exceptional divisor $p$. E_2$. We have$E_1^2 = -2$and$E_2^2 = -1$, while$E_1\cdot E_2 = 1$. Now consider$D = E_1 +2E_2$. The Zariski decomposition of$D$is$D P = (H-E) + E$0$, and $N = E_1+2E_2$, as no subdivisor is nef. Then $D\cdot E=0$E_1 = 0\$.

Post Deleted by Jack Huizenga
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