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The theory of remote sensing shows that observing a planet at multiple phaseangles ($\alpha$) is a powerful strategy to characterize its atmosphere. Here,we analyse how the information contained in reflected-starlight spectra ofexoplanets depends on the phase angle, and the potential of multi-phasemeasurements to better constrain the atmospheric properties and the planetradius ($R_p$). We simulate spectra (500-900 nm) at $\alpha$=37$^\circ$,85$^\circ$ and 123$^\circ$ with spectral resolution $R$~125-225 andsignal-to-noise ratio $S/N$=10. Assuming a H$_2$-He atmosphere, we use aseven-parameter model that includes the atmospheric methane abundance($f_{CH_4}$), the optical properties of a cloud layer and $R_p$. All theseparameters are assumed unknown a priori and explored with an MCMC retrievalmethod. We find that no single-phase observation can robustly identify whetherthe atmosphere has clouds or not. A single-phase observation at$\alpha$=123$^\circ$ and $S/N$=10 can constrain $R_p$ with a maximum error of35%, regardless of the cloud coverage. Combining small (37$^\circ$) and large(123$^\circ$) phase angles is a generally effective strategy to break multipleparameter degeneracies. This enables to determine the presence or absence of acloud and its main properties, $f_{CH_4}$ and $R_p$ in all the exploredscenarios. Other strategies, such as doubling $S/N$ to 20 for a single-phaseobservation or combining small (37$^\circ$) and moderate (85$^\circ$) phaseangles, fail to achieve this. We show that the improvements in multi-phaseretrievals are associated with the shape of the scattering phase function ofthe cloud aerosols and that the improvement is more modest forisotropically-scattering aerosols. We finally discuss that misidentifying thebackground gas in the retrievals of super-Earth observations leads to asystematic underestimate of the absorbing gas abundance.

Constraining the radius and atmospheric properties of directly imaged exoplanets through multi-phase observations