  By a theorem of Reider, a twisted bicanonical system, that means a linear
system of divisors numerically equivalent to a bicanonical divisor, on a
minimal surface of general type, is base point free if $K^2_S \geq 5$. Twisted
bicanonical systems with base points are known in literature only for
$K^2=1,2$. We prove in this paper that all surfaces in a family of surfaces
with $K^2=3$ constructed in a previous paper with G. Bini and J. Neves have a
twisted bicanonical system (different from the bicanonical system) with two
base points. We show that the map induced by the above twisted bicanonical
system is birational, and describe in detail the closure of its image and its
singular locus. Inspired by this description, we reduced the problem of
constructing a minimal surface of general type with $K^2=3$ whose bicanonical
system has base points, under some reasonable assumptions, to the problem of
constructing a curve in $\mathbb{P}^3$ with certain properties.
