We describe the phases of a solvable t -J model of electrons with infinite-range, and random, hopping, and exchange interactions, similar to those in the Sachdev-Ye-Kitaev models. The electron fractionalizes, as in an "orthogonal metal," into a fermion f , which carries both the electron spin and charge, and a boson ϕ . Both f and ϕ carry emergent Z2 gauge charges. The model has a phase in which the ϕ bosons are gapped, and the f fermions are gapless and critical, and so the electron spectral function is gapped. This phase can be considered as a toy model for the underdoped cuprates, without spatial structure. The model also has an extended, critical, "quasi-Higgs" phase where both ϕ and f are gapless, and the electron operator ∼f ϕ has a Fermi liquid-like 1 /τ propagator in imaginary time, τ . So while the electron spectral function has a Fermi liquid form, other properties are controlled by Z2 fractionalization and the anomalous exponents of the f and ϕ excitations. This quasi-Higgs phase is proposed as a toy model of the overdoped cuprates. We also describe the critical state separating these two phases.