Neutron-scattering experiments have been performed on the quasi-two-dimensional antiferromagnet ${\mathrm{K}}_{2}$${\mathrm{Mn}}_{0.978}$${\mathrm{Fe}}_{0.022}$${\mathrm{F}}_{4}$ in order to study the coexisting three- and two-dimensional ($d=3$ and $d=2$) magnetic order. The measurements were carried out in the temperature range $2 \mathrm{K}lTl60 \mathrm{K}$, and in external magnetic fields up to $H=50$ kOe applied perpendicular to the layers, i.e., parallel to the easy axis of magnetization. From temperature scans at constant field, and field scans at constant temperature, we have obtained an $H\ensuremath{-}T$ phase diagram consisting of four phases, namely, the paramagnetic $P$ phase, the antiferromagnet axial $A$ phase, an antiferromagnetic intermediate $I$ phase, and the spin-flop or planar phase. Coming from the $P$ to the $A$ phase, $d=3$ and $d=2$ ordered subsystems coexist, whereas in the $l$ phase the $d=2$ long-range order (LRO) gradually changes into the $d=3$ LRO. Upon entering the planar phase all $d=2$ LRO disappears and there is no longer a division in two subsystems. After leaving the planar phase the complete spin system remains fully $d=3$ ordered, as long as the $P$ phase is not reached. The three ordered phases are further characterized by differences in domain structures. The $H\ensuremath{-}T$ phase diagram can be explained by assuming that in this two-component antiferromagnet with competing spin anisotropies (namely, the axial dipolar anisotropy of the ${\mathrm{Mn}}^{2+}$ and ${\mathrm{Fe}}^{2+}$ ions and the planar single-ion anisotropy of the ${\mathrm{Fe}}^{2+}$ ions) at $H=0$ a mismatch occurs in the correlations along the $c$ axis between $\mathrm{xy}$ components in $d=2$ ordered clusters around the ${\mathrm{Fe}}^{2+}$ ions and the $z$ components of the $d=3$ ordered ${\mathrm{Mn}}^{2+}$ spins in adjacent layers. Applying a sufficiently strong field forces all the moments to lie in the planes and consequently the mismatch in correlation is removed. The observed $H\ensuremath{-}T$ diagram differs from that found for a weakly anisotropic Heisenberg antiferromagnet, in that the first-order spin-flop line in the latter case has become split up in two second-order transition lines, which encompass an $I$ phase, and consequently yield a tetracritical point. The $H\ensuremath{-}T$ diagram for ${\mathrm{K}}_{2}{\mathrm{Mn}}_{0.978}{\mathrm{Fe}}_{0.022}{F}_{4}$ can be explained by constructing an $x\ensuremath{-}H\ensuremath{-}T$ diagram for ${\mathrm{K}}_{2}{\mathrm{Mn}}_{1\ensuremath{-}x}{\mathrm{Fe}}_{x}{\mathrm{F}}_{4}$, using as a basis the earlier found $x\ensuremath{-}T$ diagram for $H=0$ and the $H\ensuremath{-}T$ diagram for a weakly anisotropic Heisenberg antiferromagnet ($x=0$). In the $x\ensuremath{-}H\ensuremath{-}T$ diagram a tetracritical line occurs, and the intermediate phase becomes a three-dimensional region.