While mid-latitude weather is commonly described in terms of discrete "weather systems", such as cyclones and fronts, an equivalent framework is less established in the tropics due to the wide range of relevant scales. Instead, tropical weather is often analysed either in Lagrangian terms, through the tracking of mesoscale convective systems, or via spectral decomposition into planetary-scale wave-like disturbances. Here, we use both spectral analysis and Lagrangian object tracking to develop a novel framework for analysing tropical weather systems. We apply recent theoretical advances to classify convectively coupled tropical motions into three types based on their zonal phase speed: moisture modes, inertio-gravity waves, and mixed systems. We show that, consistent with theory, the observed ratio of low-level temperature to moisture anomalies increases with the system's phase speed, although moisture variations may not be negligible for inertio-gravity waves. Moisture modes and mixed systems show distinct dynamical structures near and off the equator, while inertio-gravity waves are less clearly differentiated. These characteristics are broadly consistent across the tropics, with some regional variations. We find that moisture modes account for up to 75% of extreme rainfall events along climatological moist margins, whereas mixed systems dominate in the intertropical convergence zone. Our new tropical weather systems framework offers a new way to link large-scale tropical dynamics with precipitation.