  We consider a three-dimensional lattice $U(1) \times U(1)$ superconductor in
the London limit, with two individually conserved condensates. The problem,
generically, has two types of intercomponent interactions of different
characters. First, the condensates are interacting via a minimal coupling to
the same fluctuating gauge field. A second type of coupling is the direct
dissipationless drag represented by a local intercomponent current-current
coupling term in the free energy functional. The interplay between these two
types of interactions produces a number of physical effects not present in
previously investigated $U(1)\times U(1)$ models with only one kind of
intercomponent interaction. In this work, we present a study of the phase
diagram of a $U(1) \times U(1)$ superconductor which includes both of these
interactions. We study phase transitions and two types of competing paired
phases which occur in this general model: (i) a metallic superfluid phase
(where there is order only in the gauge invariant phase difference of the order
parameters), (ii) a composite superconducting phase where there is order in the
phase sum of the order parameters which has many properties of a
single-component superconductor but with a doubled value of electric charge. We
investigate the phase diagram with particular focus on what we call "preemptive
phase transitions". These are phase transitions {\it unique to multicomponent
condensates with competing topological objects}. A sudden proliferation of one
kind of topological defects may come about due to a fluctuating background of
topological defects in other sectors of the theory.
