Perfect tilings of 3-graphs with the generalised triangle

We establish a best-possible minimum codegree condition for the existence of a perfect tiling of a $3$-uniform hypergraph $H$ with copies of the generalised triangle $T$, which is the 3-uniform hypergraph with five vertices $a,b,c,d,e$ and three edges $abc$, $abd$, $cde$. We also give an asymptotically-optimal minimum codegree condition for the rainbow version of the problem.