Derived categories of Burniat surfaces and exceptional collections

We construct an exceptional collection $Upsilon$ of maximal possible length 6 on any of the Burniat surfaces with $K_X^2=6$, a 4-dimensional family of surfaces of general type with $p_g=q=0$. We also calculate the DG algebra of endomorphisms of this collection and show that the subcategory generated by this collection is the same for all Burniat surfaces. The semiorthogonal complement $mathcal A$ of $Upsilon$ is an almost phantom category: it has trivial Hochschild homology, and $K_0(mathcal A)=bZ_2^6$.