I'm wondering what's the grammar behind phrases like:
- We be eating
- She be sleeping
What are the grammar backgrounds for phrases in such form? Can there be any other use cases of such grammar constructions?
To use your example, with the corresponding "standard" English meaning,
Do not use this construction in "standard" English, or anywhere you may be misunderstood, because
a) it's incorrect in "standard" English
b) it may be taken as mocking how someone else speaks.
Stangdon has given one of the two possible answers. The other is that "be" here could be a use of the subjunctive - one that is rather formal and possibly dying out. (In addition to the usage discussed below, subjunctives are also used as mandatives, where they show very little sign of dying out in AmE. But I am not sure that be + a participle would ever be found in a mandative clause.)
Oxford gives this example:
Unlike rival international fairs, be they in London, New York or Maastricht, the Biennale has enormous popular appeal.
This could equally be worded as:
Unlike rival international fairs, whether they be in London, New York or Maastricht, the Biennale has enormous popular appeal.
Thus it is possible to imagine a similar example with be + participle:
Whether she be living near here or far away, I am determined to find her.
Most people would just use the indicative "is", which would be considered equally correct.