Tuning of QCD+QED simulations with C$^{\star}$ boundary conditions

We give an update on the ongoing effort of the RC$^\star$ collaboration to generate fully dynamical QCD+QED ensembles with C$^\star$ boundary conditions using the openQ$^\star$D code. The simulations were tuned to the U-symmetric point ($m_d = m_s$) with pions at $m_{\pi^{\pm}} \approx 400$ MeV. The splitting of the light mesons is used as one of three tuning observables and fixed to $m_{K^{0}} - m_{K^{\pm}} \approx 5$ MeV and $m_{K^{0}} - m_{K^{\pm}} \approx 25$ MeV on ensembles with renormalized electromagnetic coupling $\alpha_{\text{R}} \approx \alpha_{\text{phys}}$ and $\alpha_R \approx 5.5\alpha_{phys}$ respectively. The tuning of the three independent quark masses to the desired lines of constant physics is particularly challenging. We will define the chosen hadronic renormalization scheme, and we will present a tuning strategy based on a combination of mass reweighting and linear interpolation to explore the parameter space. We will comment on finite-volume effects comparing meson masses on two different volumes with $m_{\pi^{\pm}} L \approx 3.2$ and $m_{\pi^{\pm}} L \approx 5.1$. We will also provide some technical details on our updated strategy to calculate the sign of the fermionic Pfaffian, which arises in presence of C$^\star$ boundary conditions in place of the standard fermionic determinant. More technical details on the generation of the configurations can be found in J. L\"ucke's proceedings