  Flip-flop processes refer to a family of stochastic fluid processes which
converge to either a standard Brownian motion (SBM) or to a Markov modulated
Brownian motion (MMBM). In recent years, it has been shown that complex
distributional aspects of the univariate SBM and MMBM can be studied through
the limiting behaviour of flip-flop processes. Here, we construct two classes
of bivariate flip-flop processes whose marginals converge strongly to SBMs and
are dependent on each other, which we refer to as \emph{alternating}
two-dimensional Brownian motion processes}. While the limiting bivariate
processes are not Gaussian, they possess desirable qualities, such as being
tractable and having a time-varying correlation coefficient function.
