We report the first measurements of deep inelastic scattering spin-dependent azimuthal asymmetries in back-to-back dihadron electroproduction in the deep inelastic scattering process. In this reaction, two hadrons are produced in opposite hemispheres along the z axis in the virtual photon-target nucleon center-of-mass frame, with the first hadron produced in the current-fragmentation region and the second in the target-fragmentation region. The data were taken with longitudinally polarized electron beams of 10.2 and 10.6 GeV incident on an unpolarized liquid-hydrogen target using the CLAS12 spectrometer at Jefferson Lab. Observed nonzero sinΔϕ modulations in ep→e^{'}pπ^{+}X events, where Δϕ is the difference of the azimuthal angles of the proton and pion in the virtual photon and target nucleon center-of-mass frame, indicate that correlations between the spin and transverse momenta of hadrons produced in the target- and current-fragmentation regions may be significant. The measured beam-spin asymmetries provide a first access in dihadron production to a previously unexplored leading-twist spin- and transverse-momentum-dependent fracture function. The fracture functions describe the hadronization of the target remnant after the hard scattering of a virtual photon off a quark in the target particle and provide a new avenue for studying nucleonic structure and hadronization.