Is congruency same as $*$-congruency for real matrices?

Question

Two real matrices $A,B\in M_n(\mathbb R)$ are said to be congruent to each other if there exists an invertible matrix $P\in GL_n(\mathbb R)$ such that $B=P^tAP$, where $P^t$ is the transpose of $P$. Similarly, two complex matrices $A,B\in M_n(\mathbb C)$ are said to be $*$-congruent to each other if there exists an invertible matrix $P\in GL_n(\mathbb C)$ such that $B=P^{*}AP$, where $P^{*}$ is the conjugate-transpose of $P$. My question is if $A,B\in M_n(\mathbb R)$ are $*$-congruent as matrices over $\mathbb C$, must they be congruent to each other?