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I do not understand why there can be in QGIS different CRS for project and for layers. I have worked previously with subsurface petroleum software and there CRS "stuff" was lot more strict in that sense that data was almost always tied to the CRS of the project.

Will the data be prone to inconsistency if there is different CRS for project and layers?

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The project CRS is what QGIS projects your data to: the map canvas. You're basically free to choose any CRS that fits the area you are working on.

The layer CRS is the coordinate values your data is in: you can't choose that freely, because your data will be interpreted wrongly. See here for more details: https://gis.stackexchange.com/a/383437/88814

If you have set a project CRS and load data defined in another CRS, QGIS automatically transforms (reprojects) them "on the fly" (in the background, without explicit intervention) so that it fits the project CRS - at least for visualization on the map canvas.

In the background, the data are still in their original CRS. That can cause issues when using it for processing or measuring. In this case, you should consider re-projecting your data. For the visualization, most times you will not notice a difference - but see this case where there is also a visual difference.

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