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Is it possible to live with false eyes, only?

Sometimes, in a game of Go, we get a life where there is a false eye involved: There are two groups with a single real eye, each, but they are connected to one another via a false eye, allowing their individual eyes to join forces to ensure life. Something like this:

$$ .........
$$ .OOOOO...
$$ .OXXXO...
$$ .OX.XOOO.
$$ .OXXaXXO.
$$ .OOOX.XO.
$$ ...OXXXO.
$$ ...OOOOO.
$$ .........

While the false eye at a is not necessary (black could fill it and still live), it does not need to be filled either, directly providing a liberty to each of the two black chains. I wonder whether it’s possible to construct a live that entirely relies on such false eyes?

In this question, I use a purely local definition of a false eye: A false eye is a free spot that is a freedom of two distinct chains. I am aware that there are other definitions of false eyes that would call the example above a living group with three eyes.

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