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Real-analytic Jacobi forms play key roles in different areas of mathematicsand physics, but a satisfactory theory of such Jacobi forms has been lacking.In this paper, we fill this gap by introducing a space of harmonic Maass-Jacobiforms with singularities which includes the real-analytic Jacobi forms fromZwegers's PhD thesis. We provide several structure results for the space ofsuch Jacobi forms, and we employ Zwegers's $\widehat\mu$-functions to establisha theta-like decomposition.

Harmonic Maass-Jacobi forms with singularities and a theta-like decomposition