Suppose $X$ is a variety and $D$ is a Cartier divisor on $X$. Fulton argues in his Intersection theory, that if $\pi \colon \tilde X \to X$ is the blow-up of $X$ with respect to the ideal of denominators¹ of $D$, then $\pi^* D = C - E$, where $E$ is the exceptional divisor on $\tilde X$ and $C$ is an effective Cartier divisor. So after blowing-up one can write $\pi^* D$ as the difference of two effective Cartier divisors.
This left me wondering: What is an example of a Cartier divisors $D$, that cannot be written as a difference of effective divisors on $X$ without blowing-up?
¹ Locally on an affine set $U = \operatorname{Spec}(A)$, the ideal of denominators is defined as $I = \{a \in A : ad \in A\}$, where $d \in Q(A)$ is a local equation of $D$.