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").



Simple example of a perfect complex not isomorphic to a strictly perfect complex?
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?)](https://mathoverflow.net/questions/269776/can-the-homological-dimension-of-a-coherent-sheaf-explode-along-a-formal-deforma/269797#269797)   

That example is smooth, integral, and 2-dimensional over a field. However, it is not separated.


