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I believe in this paper there is an example of such group:

  • Volodia Nekrashevych,

    • Rostyslav Grigorchuk, Volodia Nekrashevych, "Amenable actions of nonamenable groups" (which can be downloaded from Volodia's webpage)

    also, this is related (but not quite clear if one can built examples that you ask from it):

    • Yair Glasner, Nicolas Monod, "Amenable actions, free products and a fixed point property"
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I believe in this paper there is an example of such group:

  • Volodia Nekrashevych, Rostyslav Grigorchuk, "Amenable actions of nonamenable groups" (which can be downloaded from Volodia's webpage)

also, this is related (but not quite clear if one can built examples that you ask from it):

  • Yair Glasner, Nicolas Monod, "Amenable actions, free products and a fixed point property"

EDIT:

In the first citation: the equivalence relation is coming from the action on the boundary of the tree satisfies all conditions that you are asking for.

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I believe in this paper there is an example of such group:

  • Volodia Nekrashevych, Rostyslav Grigorchuk, "Amenable actions of nonamenable groups" (which can be downloaded from Volodia's webpage)

also, this is related (but not quite clear if one can built examples that you ask from it):

  • Yair Glasner, Nicolas Monod, "Amenable actions, free products and a fixed point property"

EDIT:

In the first citation: the equivalence relation is coming from the action on the boundary of the tree satisfies all conditions that you are asking for.
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