  Let $\f$ be a primitive, cuspidal Hilbert modular form of parallel weight. We
investigate the Rankin convolution $L$-values $L(\f,\g,s)$, where $\g$ is a
theta-lift modular form corresponding to a finite-order character. We prove
weak forms of Kato's `false Tate curve' congruences for these values, of the
form predicted by conjectures in non-commmutative Iwasawa theory.
