  The investigation of the atmospheres of closely separated, directly imaged
gas giant exoplanets is challenging due to the presence of stellar speckles
that pollute their spectrum. To remedy this, the analysis of medium- to
high-resolution spectroscopic data via cross-correlation with spectral
templates (cross-correlation spectroscopy) is emerging as a leading technique.
We aim to define a robust Bayesian framework combining, for the first time,
three widespread direct-imaging techniques, namely photometry, low-resolution
spectroscopy, and medium-resolution cross-correlation spectroscopy in order to
derive the atmospheric properties of close-in directly imaged exoplanets. Our
framework CROCODILE (cross-correlation retrievals of directly imaged
self-luminous exoplanets) naturally combines the three techniques by adopting
adequate likelihood functions. To validate our routine, we simulated
observations of gas giants similar to the well-studied $\beta$~Pictoris~b
planet and we explored the parameter space of their atmospheres to search for
potential biases. We obtain more accurate measurements of atmospheric
properties when combining photometry, low- and medium-resolution spectroscopy
into atmospheric retrievals than when using the techniques separately as is
usually done in the literature. We find that medium-resolution ($R \approx
4000$) K-band cross-correlation spectroscopy alone is not suitable to constrain
the atmospheric properties of our synthetic datasets; however, this problem
disappears when simultaneously fitting photometry and low-resolution ($R
\approx 60$) spectroscopy between the Y and M bands. Our framework allows the
atmospheric characterisation of directly imaged exoplanets using the
high-quality spectral data that will be provided by the new generation of
instruments such as VLT/ERIS, JWST/MIRI, and ELT/METIS.
