Integrating large shares of renewables into energy systems requires energy storage technologies able to efficiently dispatch and convert different energy vectors at a local level (multi-energy storage). The paper investigates the performance of multi-energy storage based on the Thermally Integrated Pumped Energy Storage (TIPTES) technology. TIPTES stores electric energy as thermal exergy at temperatures higher or lower than the environment but also exploits additional low-temperature thermal sources, commonly from renewables or waste heat recovery, to boost the charging or discharging phases. In the paper, two different multi-energy TIPTES (mTIPTES) systems are modelled and simulated in three residential case studies where electric and thermal vectors are present, together with solar electric and thermal production. The mTIPTES annual operation is optimised via mixed-integer linear programming, linearising the operating characteristics of mTIPTES components (heat pumps, chillers, organic Rankine cycles and thermal energy storage). Finally, mTIPTES was compared against lithium-ion batteries, a benchmark for residential energy storage. The comparison is based on thermodynamic and economic figures, with the mTIPTES capital cost estimated based on cost models from the literature. The results showed that mTIPTES perform better than the battery from a thermodynamic point of view, reducing the curtailment of renewables. In economic terms, for large penetrations of renewables, this translated into an operating cost reduction of up to around 15 % compared to a system without storage, which is five percentage points lower than what was achieved with batteries. Similar results were found for CO2 emissions and primary energy consumption, for which reductions larger than batteries and up to 20 %, compared to a case without storage, can be found. Despite the operating cost reduction, the mTIPTES total annualised cost (including capital costs) is around double compared to BES, and, for the investigated configurations, mTIPTES could not achieve financial feasibility.