  In this paper we show that two dimensional (mod p) Galois representations
satisfying mild hypotheses can be lifted to p-adic Galois representations
ramified at infinitely many primes such that the characteristic polynomials of
Frobenius at a density one set of unramified primes are defined over the
rational integers. In particular we show that often one can "density one
compatibly" lift mod p and mod q Galois representations.
