  We introduce rational Dyck tilings, or $(a,b)$-Dyck tilings, and study them
by the decomposition into $(1,1)$-Dyck tilings. This decomposition allows us to
make use of combinatorial models for $(1,1)$-Dyck tilings such as the Hermite
history and the Dyck tiling strip bijection. Together with $b$-Stirling
permutations associated to the rational Dyck tilings, we obtain a
correspondence between an $(a,b)$-Dyck tiling and a tuple of $ab$ $(1,1)$-Dyck
tilings.
