  Multi-frequency matched filters (MMFs) are routinely used to detect galaxy
clusters from CMB data through the thermal Sunyaev-Zeldovich (tSZ) effect,
leading to cluster catalogues that can be used for cosmological inference. In
order to be applied, MMFs require knowledge of the cross-frequency power
spectra of the noise in the maps. This is typically estimated from the data and
taken to be equal to the power spectra of the data, assuming the contribution
from the tSZ signal of the detections to be negligible. Using both analytical
arguments and \textit{Planck}-like mock observations, we show that doing so
causes the MMF noise to be overestimated, inducing a loss of signal-to-noise.
Furthermore, the MMF cluster observable (the amplitude $\hat{y}_0$ or the
signal-to-noise $q$) does not behave as expected, which can potentially bias
cosmological inference. In particular, the observable becomes biased with
respect to its theoretical prediction and displays a variance that also differs
from its predicted value. We propose an iterative MMF (iMMF) approach designed
to mitigate these effects. In this approach, after a first standard MMF step,
the noise power spectra are reestimated by masking the detections from the
data, delivering an updated iterative cluster catalogue. Applying our iMMF to
our \textit{Planck}-like mock observations, we find that the aforementioned
effects are completely suppressed. This leads to a signal-to-noise gain
relative to the standard MMF, with more significant detections and a higher
number of them, and to a cluster observable with the expected theoretical
properties, thus eliminating any potential biases in the cosmological
constraints.
