Soliton microcombs have shown great potential in a variety of applications ranging from chip-scale frequency metrology to optical communications and photonic data center, in which light couplings among cavity transverse modes, termed as intermode interactions, are long-existing and usually give rise to localized impacts on the soliton state. Of particular interest are whispering gallery mode-based crystalline resonators, which with dense mode families, potentially feature interactions of all kinds. While effects of narrowband interactions such as spectral power spikes have been well recognized in crystalline resonators, those of broadband interactions remain unexplored. Here, we demonstrate microcombs with broadband and dispersive intermode interactions, in home-developed magnesium fluoride microresonators with an intrinsic Q-factor approaching 10 billion. In addition to conventional soliton comb generation in the single-mode pumping scheme, comb states with broadband spectral tailoring effect have been observed, via an intermode pumping scheme. Remarkably, footprints of both constructive and destructive interference on the comb spectrum have been observed, which as confirmed by simulations, are connected to the dispersive effects of the coupled mode family. Our results would not only contribute to the understanding of dissipative soliton dynamics in multi-mode or coupled resonator systems, but also extend the access to stable soliton combs in crystalline microresonators where mode control and dispersion engineering are usually challenging.