Integration of Large Language Models with search/retrieval engines has become ubiquitous, yet these systems harbor a critical vulnerability that undermines their reliability. We present the first systematic investigation of "chameleon behavior" in LLMs: their alarming tendency to shift stances when presented with contradictory questions in multi-turn conversations (especially in search-enabled LLMs). Through our novel Chameleon Benchmark Dataset, comprising 17,770 carefully crafted question-answer pairs across 1,180 multi-turn conversations spanning 12 controversial domains, we expose fundamental flaws in state-of-the-art systems. We introduce two theoretically grounded metrics: the Chameleon Score (0-1) that quantifies stance instability, and Source Re-use Rate (0-1) that measures knowledge diversity. Our rigorous evaluation of Llama-4-Maverick, GPT-4o-mini, and Gemini-2.5-Flash reveals consistent failures: all models exhibit severe chameleon behavior (scores 0.391-0.511), with GPT-4o-mini showing the worst performance. Crucially, small across-temperature variance (less than 0.004) suggests the effect is not a sampling artifact. Our analysis uncovers the mechanism: strong correlations between source re-use rate and confidence (r=0.627) and stance changes (r=0.429) are statistically significant (p less than 0.05), indicating that limited knowledge diversity makes models pathologically deferential to query framing. These findings highlight the need for comprehensive consistency evaluation before deploying LLMs in healthcare, legal, and financial systems where maintaining coherent positions across interactions is critical for reliable decision support.