The intriguing idea that strongly interacting electrons can generate spatially inhomogeneous electronic liquid-crystalline phases is over a decade old1, 2, 3, 4, 5, but these systems still represent an unexplored frontier of condensed-matter physics. One reason is that visualization of the many-body quantum states generated by the strong interactions, and of the resulting electronic phases, has not been achieved. Soft condensed-matter physics was transformed by microscopies that enabled imaging of real-space structures and patterns. A candidate technique for obtaining equivalent data in the purely electronic systems is spectroscopic imaging scanning tunnelling microscopy (SI-STM). The core challenge is to detect the tenuous but 'heavy' momentum (k)-space components of the many-body electronic state simultaneously with its real-space constituents. Sr3Ru2O7 provides a particularly exciting opportunity to address these issues. It possesses a very strongly renormalized 'heavy' d-electron Fermi liquid6, 7 and exhibits a field-induced transition to an electronic liquid-crystalline phase8, 9. Finally, as a layered compound, it can be cleaved to present an excellent surface for SI-STM.