Test-time adaptation (TTA) is a promising paradigm for handling distribution shift in time-series forecasting (TSF), where models adapt at inference time, often leveraging delayed observed data to refine predictions. In the multivariate setting, distribution shifts often exhibit cross-variate dependencies, yet existing TSF-TTA methods adapt each variate independently and ignore this cross-variate structure. Exploiting such structure motivates cross-variate interaction, but coupling variates through backbone predictions introduces direct pathways for mixing uncorrected errors across variates, a concern under the delayed supervision of TSF-TTA. We identify the \emph{interaction space} as a key design choice, and show that acting on adapter corrections that refine backbone outputs, the \emph{correction space}, rather than on the predictions themselves, avoids directly propagating backbone errors across variates. We build on this to propose \textsc{CoRe} (\textsc{Co}rrection-space Interaction \textsc{Re}finement), realizing correction-space interaction through (i) Shared-anchor Correction Refinement (SCR), which combines each variate's correction with a shared anchor through a parameter-efficient bottleneck, and (ii) input-conditioned spectral gating, which adaptively modulates the refinement from the current input window. Across seven backbones, six datasets, and four prediction horizons, \textsc{CoRe} reduces MSE by 25.82\% on average over backbones and 10.57\% over the state-of-the-art TSF-TTA method, with stronger gains at medium-to-long horizons and modest computational overhead. Data and code are available at: https://github.com/yyddou/CoReTTA

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