The real-time equation-of-motion coupled cluster (RT-EOM-CC) method has been shown to accurately predict the core and valence photoelectron spectral functions for a variety of small to moderately sized molecular systems. Previous many-body implementations included single and double CC excitations. Here, we extend the approach to include full triples CC excitations. To reduce the computational demand of these added excitations, we have implemented a more efficient approach for the time-integrator that includes an improved solver for the recursive equations and a variable time step. The new implementation is tested by computing the core spectral function of the water molecule in a reduced active space. In this space, the RT-EOM-CCSDT results agree very well with reference full configuration interaction results, up to a renormalization factor. We also compare to the experimental core and inner valence photoelectron spectra of water using a full active space, where the triple excitations fix the issues previously observed at the RT-EOM-CCSD level.