  We consider gravitational waves from a pair of monopoles or black holes that
are moving non-relativistically and are connected by a cosmic string. Shortly
after the binary's formation, the connecting string straightens due the direct
coupling of its motion to gravitational radiation. Afterwards, the motion of
the binary can be well-approximated by a non-relativistic motion of its
components that have an additional constant mutual attraction force due to the
tension of the straight string that connects them. The orbit shrinks due to the
gravitational radiation backreacting on the binary's components. We find that
if the binary's semimajor axis $a\gg \sqrt{R_1 R_2/{\mu}}$, its eccentricity
grows on the inspiral's timescale; here $R_1$ and $R_2$ are the gravitational
radii of the binary components, and $\mu$ is the dimensionless tension of the
string. When the eccentricity is high, it approaches unity super-exponentially.
If the binary's components are monopole-antimonopole pair, this leads to the
physical collision that would likely destroy the string and annihilate the
monopoles when the semimajor axis is still many orders of magnitude greater
than the string thickness. If the binary's components are black holes, then the
eccentricity reaches its peak when $a\sim \sqrt{R_1 R_2/\mu}$, and then decays
according to the standard Peter's formula. The black-hole spins initially
become locked to the orbital motion, but then lag behind as the inspiral
proceeds. We estimate the string-tension-induced dimensionless spins just prior
to the merger and find them to be $\sim\mu^{3/8}\ll 1$.
