  A totally asymmetric exclusion process on a ring with $\nu$ non-conserved
internal degrees of freedom, where particles hop forward with a rate that
depends on their internal state, has been studied. We show, using a mapping of
the model to a zero range process with $\nu$ different kinds of boxes, that
steady state weights can be written in a matrix product form and calculate the
spatial correlations exactly. A comparison of the model with an equivalent
conserved system reveals that unequal hopping rates of particles belonging to
different internal states is responsible for the non-trivial correlations.
