  In order to enable semiconductor-based quantum computing with many qubits,
issues like residual interqubit coupling and constraints from scalable control
hardware need to be tackled to retain the high gate fidelities demonstrated in
current single-qubit devices. Here, we focus on two exchange-coupled
singlet-triplet spin qubits, considering realistic control hardware as well as
Coulomb and exchange coupling that cannot be fully turned off. Using measured
noise spectra, we optimize realistic control pulses and show that two-qubit
(single-qubit) gate fidelities of 99.90\% ($\ge 99.69\%$) can be reached in
GaAs, while 99.99\% ($\ge 99.95\%$) can be achieved in Si.
