  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 $K_0(\mathcal
A)=\bZ_2^6$.
