  Suppose $\rho_1, \rho_2$ are two $\ell$-adic Galois representations of the
absolute Galois group of a number field, such that the algebraic monodromy
group of one of the representations is connected and the representations are
locally potentially equivalent at a set of places of positive upper density. We
classify such pairs of representations and show that up to twisting by some
representation, it is given by a pair of representations one of which is
trivial and the other abelian.
  Consequently, assuming that the first representation has connected algebraic
monodromy group, we obtain that the representations are potentially equivalent,
provided one of the following conditions hold: (a) the first representation is
absolutely irreducible; (b) the ranks of the algebraic monodromy groups are
equal; (c) the algebraic monodromy group of the second representation is also
connected and (d) the commutant of the image of the second representation
remains the same upon restriction to subgroups of finite index of the Galois
group.
