The specific Illusory Reality duration beats the general spell duration.
The general rule in D&D 5e is that "specific beats general". So while in general an illusion created by a spell lasts exactly as long as the spell lasts, the Illusory Reality class feature has a specific statement about the duration of its effects.
This means that if an illusionist casts a major image of a blacksmith's forge, with a duration of 10 minutes, they can use Illusory Reality to make the anvil from the forge real. If they do this up to nine minutes after casting the major image, after one minute the anvil will stop being real and revert to being an illusion for the remainder of the spell's duration.
If the illusionist uses Illusory Reality immediately before major image ends (because of its duration, or because the illusionist willingly or unwillingly dropped concentration on the spell), the anvil will remain real while the rest of the illusion fades away. After the one minute of reality afforded to the anvil by Illusory reality, it will neither real nor illusory, and thus cease to exist.