  In strongly interacting systems with multiple energy bands, the interplay
between electrons with different effective masses and the enlarged Hilbert
space drives intricate correlated phenomena that do not occur in single-band
systems. Recently, magic-angle twisted trilayer graphene (MATTG) has emerged as
a promising tunable platform for such investigations: the system hosts both
slowly dispersing, "heavy" electrons inhabiting its flat bands as well as
delocalized "light" bands that disperse as free Dirac fermions. Most
remarkably, superconductivity in twisted trilayer graphene and multilayer
analogues with additional dispersive bands exhibits Pauli limit violation and
spans a wider range of phase space compared to that in twisted bilayer
graphene, where the dispersive bands are absent. This suggests that the
interactions between different bands may play a fundamental role in stabilizing
correlated phases in twisted graphene multilayers. Here, we elucidate the
interplay between the light and heavy electrons in MATTG as a function of
doping and magnetic field by performing local compressibility measurements with
a scanning single-electron-transistor microscope. We establish that commonly
observed resistive features near moir\'e band fillings $\nu$=-2, 1, 2 and 3
host a finite population of light Dirac electrons at the Fermi level despite a
gap opening in the flat band sector. At higher magnetic field and near charge
neutrality, we discover a new type of phase transition sequence that is robust
over nearly 10 micrometers but exhibits complex spatial dependence. Mean-field
calculations establish that these transitions arise from the competing
population of the two subsystems and that the Dirac sector can be viewed as a
new flavor analogous to the spin and valley degrees of freedom.
