Despite growing interest in applications based on natural customer support conversations,there exist remarkably few publicly available datasets that reflect the expected characteristics of conversations in these settings. Existing task-oriented dialogue datasets, which were collected to benchmark dialogue systems mainly in written human-to-bot settings, are not representative of real customer support conversations and do not provide realistic benchmarks for systems that are applied to natural data. To address this gap, we introduce NatCS, a multi-domain collection of spoken customer service conversations. We describe our process for collecting synthetic conversations between customers and agents based on natural language phenomena observed in real conversations. Compared to previous dialogue datasets, the conversations collected with our approach are more representative of real human-to-human conversations along multiple metrics. Finally, we demonstrate potential uses of NatCS, including dialogue act classification and intent induction from conversations as potential applications, showing that dialogue act annotations in NatCS provide more effective training data for modeling real conversations compared to existing synthetic written datasets. We publicly release NatCS to facilitate research in natural dialog systems