Even though iterative solvers like the Conjugate Gradients method (CG) have been studied for over fifty years, fault tolerance for such solvers has seen much attention in recent years. For iterative solvers, two major reliable strategies of recovery exist: checkpoint-restart for backward recovery, or some type of redundancy technique for forward recovery. Important redundancy techniques like ABFT techniques for sparse matrix-vector products (SpMxV) have recently been proposed, which increase the resilience of CG methods. These techniques offer limited recovery options, and introduce a tolerable overhead. In this work, we study a more powerful resilience concept, which is redundant multithreading. It offers more generic and stronger recovery guarantees, including any soft faults in CG iterations (among others covering ABFT SpMxV), but also requires more resources. We carefully study this redundancy/efficiency conflict. We propose a fault tolerant CG method, called TwinCG, which introduces minimal wallclock time overhead, and significant advantages in detection and correction strategies. Our method uses Dual Modular Redundancy instead of the more expensive Triple Modular Redundancy; still, it retains the TMR advantages of fault correction. We describe, implement, and benchmark our iterative solver, and compare it in terms of efficiency and fault tolerance capabilities to state-of-the-art techniques. We find that before parallelization, TwinCG introduces around 5-6% runtime overhead compared to standard CG, and after parallelization efficiently uses BLAS. In the presence of faults, it reliably performs forward recovery for a range of problems, outperforming SpMxV ABFT solutions.