Style features such as friendly, helpful, or concise are widely used in prompts to steer the behavior of Large Language Model (LLM) conversational agents, yet their unintended side effects remain poorly understood. In this work, we present the first systematic study of cross-feature stylistic side effects. We conduct a comprehensive survey of 127 conversational agent papers from ACL Anthology and identify 12 frequently used style features. Using controlled, synthetic dialogues across task-oriented and open domain settings, we quantify how prompting for one style feature causally affects others via a pairwise LLM as a Judge evaluation framework. Our results reveal consistent and structured side effects, such as prompting for conciseness significantly reduces perceived expertise. They demonstrate that style features are deeply entangled rather than orthogonal. To support future research, we introduce CASSE (Conversational Agent Stylistic Side Effects), a dataset capturing these complex interactions. We further evaluate prompt based and activation steering based mitigation strategies and find that while they can partially restore suppressed traits, they often degrade the primary intended style. These findings challenge the assumption of faithful style control in LLMs and highlight the need for multi-objective and more principled approaches to safe, targeted stylistic steering in conversational agents.