  We discuss exact scalar field solutions describing gravitating compact
objects in the Eddington-inspired Born-Infeld gravity (EiBI), a member of the
class of (metric-affine formulated) Ricci-based gravity theories (RBGs). We
include a detailed account of the RBGs/GR correspondence exploited to
analytically solve the field equations. The single parameter $\e$ of the EiBI
model defines two branches for the solution. The $\e>0$ branch may be described
as a `shell with no interior', and constitutes an ill-defined, geodesically
incomplete spacetime. The more interesting $\e<0$ branch admits the
interpretation of a `wormhole membrane', an exotic horizonless compact object
with the ability to transfer particles and light from any point on its surface
(located slightly below the would-be Schwarzschild radius) to its antipodal
point, in a vanishing fraction of proper time. This is a single example
illustrating how the structural modifications introduced by the metric-affine
formulation may lead to significant departures from GR even at astrophysically
relevant scales, giving rise to physically plausible objects radically
different from those we are used to think of in the metric approach, and that
could act as a black hole mimickers whose shadows might present distinguishable
signals.
