Customfields belonging to specific projects are shown when those projects are selected

I have created 2 customfields, each one belonging to a specific project.
When I perform a search on those 2 projects, I can see the 2 customfields in the search form.
The problem is that if I select a value for customfield number 1, the search returns results in the project number 1 only. I have no results for the project number 2.
In addition, if I select a value for Customfield number 1 and a value for Customfield number 2, I have 0 result.

IMHO:

When selecting multiple projects, customfields not being configured for the context (ie available for the 2 projects) shouldn't be visible in the search form.

OR

When performing the search, Jira should search results with a "OR" condition whereas a "AND" condition: search issues in project 1, project2 where customfield1=value or customfield2=value
In this case, we will have results.

Thanks,
Cécile