  We propose a new platform for observing the Josephson diode effect: the
Andreev molecule. This nonlocal electronic state is hosted in circuits made of
two closely spaced Josephson junctions, through the hybridization of the
Andreev states. The Josephson diode effect occurs at the level of one
individual junction while the other one generates the required time-reversal
and spatial-inversion symmetry breaking. We present a microscopic description
of this phenomenon based on fermionic Andreev states, focusing on single
channels in the short limit, and we compute both supercurrent and energy
spectra. We demonstrate that the diode efficiency can be tuned by magnetic flux
and the junctions transmissions, and can reach $45~\%$. Going further, by
analyzing the Andreev spectra, we demonstrate the key role played by the
continuum, which consists of leaky Andreev states and is largely responsible
for the critical current asymmetry. On top of proposing an experimentally
accessible platform, this work elucidates the microscopic origin of the
Josephson diode effect at the level of the fermionic Andreev states.
