We show that every probability-measure-preserving action of a countable amenable group G can be tiled, modulo a null set, using finitely many finite subsets of G ("shapes'') with prescribed approximate invariance so that the collection of tiling centers for each shape is Borel. This is a dynamical version of the Downarowicz--Huczek--Zhang tiling theorem for countable amenable groups and strengthens the Ornstein--Weiss quasi-tiling lemma. This work is joint with Conley--Jackson--Kerr--Marks--Tucker-Drob and is motivated by the long standing open question asking whether the equivalence relation generated by a Borel action of a countable amenable group must always be hyperfinite.