Several therapy routines require deep breathing exercises as a key component and patients undergoing such therapies must perform these exercises regularly. Assessing the outcome of a therapy and tailoring its course necessitates monitoring a patient's compliance with the therapy. While therapy compliance monitoring is routine in a clinical environment, it is challenging to do in an at-home setting. This is so because a home setting lacks access to specialized equipment and skilled professionals needed to effectively monitor the performance of a therapy routine by a patient. For some types of therapies, these challenges can be addressed with the use of consumer-grade hardware, such as earphones and smartphones, as practical solutions. To accurately monitor breathing exercises using wireless earphones, this paper proposes a framework that has the potential for assessing a patient's compliance with an at-home therapy. The proposed system performs real-time detection of breathing phases and channels with high accuracy by processing a $\mathbf{500}$ ms audio signal through two convolutional neural networks. The first network, called a channel classifier, distinguishes between nasal and oral breathing, and a pause. The second network, called a phase classifier, determines whether the audio segment is from inhalation or exhalation. According to $k$-fold cross-validation, the channel and phase classifiers achieved a maximum F1 score of $\mathbf{97.99\%}$ and $\mathbf{89.46\%}$, respectively. The results demonstrate the potential of using commodity earphones for real-time breathing channel and phase detection for breathing therapy compliance monitoring.