We propose a two parameters extension of the flat $\mathrm{\Lambda }\mathrm{CDM}$ model to capture the impact of matter inhomogeneities on our cosmological inference. Non virialized but nonlinearly evolving overdense and underdense regions, whose abundance is quantified using the Press-Schechter formalism, are collectively described by two effective perfect fluids ${\rho }_{\mathrm{c}}$, ${\rho }_{\mathrm{v}}$ with nonvanishing equation of state parameters $w_{\mathrm{c},\mathrm{v}}\ne 0$. These fluids are coupled to the pressureless dust, akin to an interacting DM-DE scenario. The resulting phenomenology is very rich, and could potentially address a number of inconsistencies of the standard model, including a simultaneous resolution of the Hubble and ${\sigma }_8$ tensions. To assess the viability of the model, we set initial conditions compatible to the Planck 2018 best fit $\mathrm{\Lambda }\mathrm{CDM}$ cosmology and fit its additional parameters using SN Ia observations from DESY5, BAO distances from DESI DR2 and a sample of uncorrelated $f{\sigma }_8$ measurements. Our findings show that backreaction effects from the cosmic web could restore the concordance between early and late Universe cosmological probes.

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• Cosmology
• Evolution of the Universe
• Large scale structure of the Universe
• Relativistic aspects of cosmology