FATMAP difficulty grade
The Rimpfischhorn is a remote peak from any direction, also often climbed from the Zermatt side, but here described from the Britannia Hut, which you can also base yourself at for the Strahlhorn.
The start is in common for both peaks.
From the hut follow the path descending SW beneath the rocky buttress of the Hinter Allalin ridge to get onto the Hohlaub Glacier.
Go SSW across the Hohlaub Glacier passing under point 3150m to gain the Allalin Glacier.
Follow the R side of the glacier SW under the large cliffs to your R, negotiating your way round the crevasses, before the angle eases off around 3250m.
At this point your route deviates from the route up the Strahlhorn, as you angle up to the R, more steeply, taking the glacial slopes leading to the Allalinpass at 3556m.
From here you head SW, passing through, at 3650m, or under an obvious rocky spur, followed by passing beneath another similar spur, further to the S.
The route then steepens slightly, weaving through complicated glacial terrain, heading SE then S, arriving at the Rimpfischsattel at 3989m.
To your R is the winter ski summit at 4009m, your true summit above to the L.
The route becomes mixed from here taking a gully on the R which leads to the summit ridge.
There are odd bits of fixed protection of variable quality both in the gully and on the ridge, leading to your summit cross at 4199m.
Descent is the reverse of ascent, taking care not to knock anything down on others climbing up.
The route is marked reascending to the Britannia Hut, but you can clearly head down to Saas, by lift or descend on the Zermatt side towards the Täsch Hut.