  In this paper, we describe the automorphic properties of the Fourier
coefficients of meromorphic Jacobi forms. Extending results of Dabholkar,
Murthy, and Zagier, and Bringmann and Folsom, we prove that the canonical
Fourier coefficients of a meromorphic Jacobi form $\phi(z;\tau)$ are the
holomorphic parts of some (vector-valued) almost harmonic Maass forms. We also
give a precise description of their completions, which turn out to be uniquely
determined by the Laurent coefficients of $\phi$ at each pole, as well as some
well known real analytic functions, that appear for instance in the completion
of Appell-Lerch sums.
