What is the difference between:
The company would rather each employee be provided with ID card.
The company would rather each employee were provided with ID card.
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Here is my understanding of the way they are supposed to be used:
The difference is between the present subjunctive (also called the mandative subjunctive) and the past subjective (also called the irrealis).
The company would rather (that) each employee be provided with an ID card.
This is the mandative subjunctive, generally used for orders and suggestions. To me, this sounds like the company is trying to decide between two (or more) possible policies, and they prefer this one. It still might not happen, if other considerations prevent it (i.e., maybe it's too expensive).
The company would rather (that) each employee were provided with an ID card.
This is the irrealis mood. It is used for hypothetical situations. Here, it is not the case that each employee has been provided with an ID card, and the company wishes that it were the case. Unlike the first sentence, there is no implication that they are considering taking action to make it the case, although this possibility isn't ruled out.
Searching with Google, it seems that there is not that sharp a distinction between these in actual use. While the above difference seems to describe the most probable meanings, you can find a number of cases where be is used for what are clearly hypothetical or impossible situations, and a number of cases where were is used for what look like actual suggestions.
The company would rather each employee were provided with an ID card. /
The company would rather each employee were in possession of an ID card.
= The company would prefer it if each employee had an ID card. [statement of what the company would like the present set-up to be, not what it actually is]
The company would rather each employee be provided with an ID card.
= The company wants each employee to be given an ID card.
[statement of the company's action plan]
Both boil down to the same thing, really.