Prokaryotic anti-phage immune systems use TIR (toll/interleukin-1 receptor) and cGAS (cyclic GMP-AMP synthase) enzymes to produce 199-39/199-29 glycocyclic ADPR (gcADPR) and cyclid di-/tri-nucleotides (CDNs and CTNs) signaling molecules that limit phage replication, respectively. However, how phages neutralize these common systems is largely unknown. Here, we show that Thoeris anti-defense proteins Tad1 and Tad2 both have anti-CBASS activity by simultaneously sequestering CBASS cyclic oligonucleotides. Strikingly, apart from binding Thoeris signals 199-39 and 199-29 gcADPR, Tad1 also binds numerous CBASS CDNs/CTNs with high affinity, inhibiting CBASS systems using these molecules in vivo and in vitro. The hexameric Tad1 has six binding sites for CDNs or gcADPR, which are independent from two high affinity binding sites for CTNs. Tad2 also sequesters various CDNs in addition to gcADPR molecules, inhibiting CBASS systems using these CDNs. However, the binding pockets for CDNs and gcADPR are different in Tad2, whereby a tetramer can bind two CDNs and two gcADPR molecules simultaneously. Taken together, Tad1 and Tad2 are both two-pronged inhibitors that, alongside anti-CBASS protein 2, establish a paradigm of phage proteins that flexibly sequester a remarkable breadth of cyclic nucleotides involved in TIR- and cGAS-based anti-phage immunity.