  We use extensive Monte Carlo transfer matrix calculations on infinite strips
of widths $L$ up to 30 lattice spacing and a finite-size scaling analysis to
obtain critical exponents and conformal anomaly number $c$ for the
two-dimensional $XY$-Ising model. This model is expected to describe the
critical behavior of a class of systems with simultaneous $U(1)$ and $Z_2$
symmetries of which the fully frustrated $XY$ model is a special case. The
effective values obtained for $c$ show a significant decrease with $L$ at
different points along the line where the transition to the ordered phase takes
place in a single transition. Extrapolations based on power-law corrections
give values consistent with $c=3/2$ although larger values can not be ruled
out. Critical exponents are obtained more accurately and are consistent with
previous Monte Carlo simulations suggesting new critical behavior and with
recent calculations for the frustrated $XY$ model.
