Simple example of a perfect complex not isomorphic to a strictly perfect complex?

Question

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).

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Definitions:

1. A strictly perfect complex is an object in the derived category $D(X)$ isomorphic to a bounded complex of vector bundles
2. 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").

Answer 1

Perhaps it is silly to post this as an answer. My answer to your other question also answers this question.
Can the homological dimension of a coherent sheaf explode along a formal deformation? (is the resolution property hereditary for formal deformations?)
That example is smooth, integral, and 2-dimensional over a field. However, it is not separated.