diff --git a/src/factorizations/truncation.jl b/src/factorizations/truncation.jl
index b9d060fec..3bc1ebf48 100644
--- a/src/factorizations/truncation.jl
+++ b/src/factorizations/truncation.jl
@@ -191,6 +191,23 @@ function _sort_and_perm(values::SectorVector; by = identity, rev::Bool = false)
     return values_sorted, perms
 end
 
+function _findtruncvalue_order(values::SectorVector, n::Int; by = identity, rev::Bool = false)
+    I = sectortype(values)
+    p = sortperm(parent(values); by, rev)
+
+    if FusionStyle(I) isa UniqueFusion # dimensions are all 1
+        return n <= 0 ? nothing : p[min(n, length(p))]
+    else
+        dims = similar(values, Base.promote_op(dim, I))
+        for (c, v) in pairs(dims)
+            fill!(v, dim(c))
+        end
+        cumulative_dim = cumsum(Base.permute!(parent(dims), p))
+        k = findlast(<=(n), cumulative_dim)
+        return isnothing(k) ? k : p[k]
+    end
+end
+
 # findtruncated
 # -------------
 # Generic fallback
@@ -202,25 +219,25 @@ function MAK.findtruncated(values::SectorVector, ::NoTruncation)
     return SectorDict(c => Colon() for c in keys(values))
 end
 
+# TruncationByOrder strategy:
+# - find the howmany'th value of the input sorted according to the strategy
+# - discard everything that is ordered after that value
+
 function MAK.findtruncated(values::SectorVector, strategy::TruncationByOrder)
-    values_sorted, perms = _sort_and_perm(values; strategy.by, strategy.rev)
-    inds = MAK.findtruncated_svd(values_sorted, truncrank(strategy.howmany))
-    return SectorDict(c => perms[c][I] for (c, I) in inds)
-end
-function MAK.findtruncated_svd(values::SectorVector, strategy::TruncationByOrder)
-    I = keytype(values)
-    truncdim = SectorDict{I, Int}(c => length(d) for (c, d) in pairs(values))
-    totaldim = sum(dim(c) * d for (c, d) in truncdim; init = 0)
-    while totaldim > strategy.howmany
-        next = _findnexttruncvalue(values, truncdim; strategy.by, strategy.rev)
-        isnothing(next) && break
-        _, cmin = next
-        truncdim[cmin] -= 1
-        totaldim -= dim(cmin)
-        truncdim[cmin] == 0 && delete!(truncdim, cmin)
+    k = _findtruncvalue_order(values, strategy.howmany; strategy.by, strategy.rev)
+
+    if isnothing(k)
+        # discard everything
+        return SectorDict{sectortype(values), UnitRange{Int}}()
+    else
+        val = strategy.by(values[k])
+        strategy = trunctol(; atol = val, strategy.by, keep_below = !strategy.rev)
+        return MAK.findtruncated_svd(values, strategy)
     end
-    return SectorDict(c => Base.OneTo(d) for (c, d) in truncdim)
 end
+# disambiguate
+MAK.findtruncated_svd(values::SectorVector, strategy::TruncationByOrder) =
+    MAK.findtruncated(values, strategy)
 
 function MAK.findtruncated(values::SectorVector, strategy::TruncationByFilter)
     return SectorDict(c => findall(strategy.filter, d) for (c, d) in pairs(values))