It has been shown that the coherent detection of long range (LoRa) signals only provides marginal gains of around 0.7 dB on the additive white Gaussian noise (AWGN) channel. However, ALOHA-based massive Internet-of-Things systems, including LoRa, often operate in the interference-limited regime. Therefore, in this work, we examine the performance of the LoRa modulation with coherent detection in the presence of interference from another LoRa user with the same spreading factor. We derive rigorous symbol- and frame error rate (FER) expressions as well as bounds and approximations for evaluating the error rates. The error rates predicted by these approximations are compared against error rates found by Monte Carlo simulations and shown to be very accurate. We also compare the performance of LoRa with coherent and noncoherent receivers and we show that the coherent detection of LoRa is significantly more beneficial in interference scenarios than in the presence of only AWGN. For example, we show that coherent detection leads to a 2.5-dB gain over the standard noncoherent detection for a signal-to-interference ratio (SIR) of 3 dB and up to a 10-dB gain for an SIR of 0 dB. Moreover, we show that with coherent detection it is easier to obtain useful and relevant FER values even for negative SIR values.