Exceptional collections on nonminimal Enriques surfaces

By Orlov’s formula, the derived category of blow up X = \operatorname {Bl}_pX’ \to X’ contains \operatorname {D}^{\mathsf {b}}(X’) as a semiorthogonal component. This raises an interesting question: does there exist a variety X’ such that \operatorname {D}^{\mathsf {b}}(X’) does not admit an exceptional collection of maximal length, but \operatorname {D}^{\mathsf {b}}(X) admits an exceptional collection of maximal length? We give such an example when X’ is a minimal Enriques surface.