I'm looking for the simplest possible example (one that's easy to remember) for the situation described in the title. More precisely I'm looking for the following example:

A (probably has to be singular) algebraic surface $X$ (2-dimensional, reduced, integral, finite type over $k$ algebraically closed) and a complex of coherent sheaves $\mathcal{F}^\bullet$ on $X$ whose image in $D(X)$ is perfect but not quasi-isomorphic to a strictly perfect complex. (Bonus points for a simple toric surface example).

Definitions:

- A
**strictly perfect**complex is an object in the derived category $D(X)$ isomorphic to a bounded complex of vector bundles - A
**perfect complex**is an object in the derived category $M \in D(X)$ satisfying that for every point $x \in X$ there's a open zariski neighborhood $x \in U$ over which $M_U \in D(U)$ is perfect ("locally strictly perfect").